In its VI Congress of 1928, either from a lack of proper information, incompetent analysis from the CPs or from the Comintern itself, a number of countries (Spain, Portugal, Poland, Hungary, the Balkan countries, etc.) were defined as "having numerous survivals of semi-feudal relationships in agriculture, possessing, to a certain extent, the material pre-requisites for socialist construction, and in which the bourgeois-democratic reforms have not yet been completed". By haphazardly applying the lessons of the February and October revolutions (bourgeois-democratic and proletarian revolutions respectively) to these other countries supposedly not yet fully consolidated as parts of the imperialist pyramid, the national liberation theory for colonies prioritizing alliances with national-revolutionary bourgeoisies was transplanted to fit into these capitalist (and some already fully imperialist, such as Portugal and Spain) countries.
The strategy of social alliances that characterized the united front strategy from the III Congress was morphed into one of political alliances with certain parts of the bourgeoisie. Instead of properly characterizing the bourgeoisie in these countries as being broadly split between a liberal and a socialdemocratic end, it was instead seen as being split between a reactionary and a progressive bourgeoisie. The latter was to be allied with, to finish the tasks of the bourgeois-democratic revolutions in these countries to enable the possibility of a proletarian revolution. It was thought that if the CPs participated in these bourgeois-democratic revolutions, they'd gain hegemony within the working class as soon as the progressive bourgeoisie halted all "progressive" goals upon reaching power, in a situation similar to the bolshevik experience of the February Revolution.
In countries such as Spain, which encompassed multiple nations, this progressive bourgeoisie was identified with the bourgeoisies of those stateless nations, further degrading the capacity of the CPs to lead the revolutionary organization of the working class, although it became more relevant way after the Comintern's dissolution.
This laid the foundations for the popular front strategy, evidenced by the fact that some CPs began to create popular fronts on the basis of the programme of the VI Congress, before popular fronts were officially adopted in the VII Congress, in 1935.
The popular front as a strategy was, as said before, adopted officially by the VII and final Congress of the Comintern. The united front strategy from the III Congress placed the priority on a class unity, the unity of the working class under a single Communist Party. In the wake of fascist triumphs across Europe and what seemed like an inability from the CPs to oppose fascists, they restructured the priorities of the International. Instead of a proletarian revolution, to be carried out by a CP and the working class it unites plus the convenient social alliances, the priority became the defeat of organized fascism through any means, including the collaboration with social-democracy, the parties of the II International. This time, class unity was transplanted to refer to the alliance between the working class and its ultimately counter-revolutionary foes. The social-democratic parties that hadn't yet capitulated to fascism like the SPD did, were differentiated with what the International referred to as "social-fascists". I won't waste too many lines explaining the evident historical failure of the popular front strategy in every single context it was applied to. The CP of Spain, had it maintained the united front strategy and had it properly applied the lessons of the 1934 revolution, would have probably been able to sieze power by the time of the reactionary coup of 1936.
The legacy (or rather, damage) of those analyses is not confined to their historical movement and were reinforced after the change in the USSR's course after Stalin's death and his demonization. The USSR had been, for better or for worse, a referent for CPs worldwide, and the change in course left many CPs even more isolated, especially those in the global south, many of which still claimed Stalin's legacy because of the internationalism crucial to the development of these CPs that stopped after his death. Having identified certain countries as needing a finished bourgeoisie-democratic revolutions before enabling the possibility of a proletarian revolution, and having muddled the waters as to what class unity meant, the infiltration of petit-bourgeois and worker-aristocratic tendencies into the worker's movement was massively facilitated.
The so-called "national pathways" to socialism became a common justification, born out of the mischaracterization of the stage of capitalist development in the VI Congress of the CI, for many deviations. Politically, it always lead to the collaboration between the CP and reformist powers in their bourgeois governments (aided by the memory of popular-frontism), such as in the case of Italy's and Spain's CP, an abdication of the historical call to the organization of the proletarian revolution. Organizationally, it often metastazised in the degradation of discipline in the party and the repudiation of the workplace as the territorial basis of party organization (one which had admittedly low implementation in the times of the CI, but which, however, is a direct manifestation of recognizing the working class as the prime revolutionary class of our time, and did not lead to outright reformist drift). Curiously enough, these "national pathways", against the supposedly dogmatic and irrelevant leninist principles, always arrived at the same place; integration into bourgeois governments and the eventual complete dissolution of explicit revolutionary goals and strategy.
The mischaracterization of the national question in the countries with a "medium development of capitalism" eventually evolved into the wholesale attribution of a progressive character to separatist movements in Europe, related with the rhethoric of the national pathways to socialism. Meaning that the nationalist and separatist movements of Europe, with no other program than a mere reconfiguration of the borders along which the capitalist and working classes are divided, were seen as movements whose support was a necessary step towards socialism. Movements such as ETA (that had an explicitly marxist component during a certain period) and the IRA.
These errors are not solely culpable and I don't think it's useful to wallow about them or wonder what could have been had they not been committed. After all, "it is not the consciousness of men that determines their existence, but their social existence that determines their consciousness". These errors became opportunities for the worker aristocracy, as the right hand man of the bourgeoisie in the worker's movement, to promote their petit-bourgeois aspirational ideological deviations. These deviations were not only political, they also manifested in terms of organizational structure of the CPs, either in advance to or in lockstep with the political deviations.
It's most visible in the questions related to the bolshevization of the Party; in regards to to the basic organization of the Party and the requisites for membership.
For some context, the Comintern established the factory or company cell as the basic structure of the Party in its III Congress (1921), because it's what's most articulated for political activity in the setting of the development of the mode of production itself, the main contradiction and the source of the class struggle. This wasn't really applied or even discussed in most parties until the meeting of the Comintern's CC of January 1924 (and the organization conferences of 1925 and 1926), where they identify the KPD's disregard of the factory cell, hence its distancing from the workers, as one of the main factors contributing to the defeat of the 1923 Hamburg uprising. This analysis is also proof that the common criticism that the cell structure is wholly based on the Tsarist empire's conditions is false, and that it was arrived at from the experiences of many more CPs.
By 1929, the KPD had 14'6% of its militants in factory cells, 45% in street cells, and 40% in radios (immediately superior structure to the cell) without a cell. There was a similar situation, for instance, in the French and Czechoslovakian CPs. This was due to many reasons, among which was a failure to understand the figure of a militant (more on that later) and resistance to the bolshevization process. Amadeo Bordiga, representing a minority in the Italian CP (PCI) and in a historical echo, argued that the cell structure was only appropriate for the Russian context, and that Italy needed its own (national) path.
Loriot, from the French CP, argued that the radios should have political sovereignty, because factory/company cells had no capacity for political education, completely disregarding the role of political practice in education and misunderstanding the roles of a cell defined by the Comintern.
The Czechoslovak CP (KSC), originally formed from mostly social-democratic parties (first a majority of the Czechoslovak Social-Democratic Party (CSDSD) in 1920, then the left wing of the Slovak Social-Democratic Party (SDSS) and the social-democrats of Subcarpathia in 1921, the last two being briefly joined into the "Marxist Left in Slovakia and the Transcarpathian Ukraine"), was extremely reticent of giving up the social-democratic conception of separating political (electoral) work and economic, immediate work between the party and the union respectively. This was one of the causes for the crisis and eventual rupture of the party between 1927 and 1929. By 1930, of the party's 57% worker membership, 14% (of the total) were in factory cells.
As a continuity of the previously discussed political mistakes of the Comintern, the Italian CP during the end of and after WW2 moved away from the bolshevik Party model and towards a mass model, the Partito Nuovo, justified in 1944 with the idea of the Single Working Class Party (union of all "proletarian" parties, that is, communist and social-democratic), itself developed from the popular front strategy. This was paired with a political shift, away from the socialist revolution and towards the construction of a democracy, without class character. This is a legacy of the Comintern's VI Congress, discussed above. The factory cell was also questioned, stressing in 1957 that the factory cell militants should also be active in their corresponding street cells, to be eventually broadened and removed. The PCI also disregarded the search for communist hegemony or even relevancy in the unions, alleging that there was no longer a worker aristocracy to influence the unions towards bourgeois positions.
The Spanish CP cheapened the militancy figure, in a clear move away from a revolutionary vanguard and towards a struggle within a spanish democracy, following the national reconciliation policy, itself a logical conclusion of the assessment of Spain as a country with a bad enough capitalist development to not be able to have a proletarian revolution. I could go into detail about the timeline, but I don't think it's necessary. The eurocommunist and similar deviations in the late 20th century of the West European CPs was a destination which, in retrospect, was firmly in the straight path forward for those two political mistakes of the Comintern.
Outside Europe, Browder's decision to dissolve the CPUSA in 1944, justified by the dissolution of the Comintern, the Tehran conference, and based on the National Unity policy as a continuation of the Popular Front. Eventually, it was this distancing from the Comintern and destruction of the CPUSA that allowed for opportunist, worker-aristocratic deviations such as patriotic socialism to flourish. This had a devastating effect not only on the USA, but also in Latin America. Some of these effects were name changes and the abandonment of factory cells, such as in the Mexican CP (PCM).
Overall, this has been the effect of the influence of petit-bourgeois, worker-aristocratic and generally left and right deviations in the development of the Third International's Communist Parties. The opposition to these mistakes, their direct and secondary effects is essential for each country's Communist Party, capitalized, to come about again, on the basis of proletarian internationalism and the proven —directly and by failure of alternatives— bolshevik organization of the CP.