I'm going to stick with the Italian flavour of filming without a soundtrack since director Federico Fellini used it and information is more readily available on one celebrated international director than a dozen others with less fame.
Filming without a recorded soundtrack was common among Italian filmmakers. Although I have no sourced proof, I strongly suspect that this was done to increase the pool of available on-screen talent since filming without sound did not require any one actor to speak a particular language. Additionally, multiple voice-over soundtracks could be cheaply produced in a variety of languages for international release.
However, with regard to the referenced example La Strada, director Federico Fellini did not have the actors speak their lines in his/her native language. Rather, a system known as 'number system' or 'numerological diction' was employed.
The actors would recite a series of numbers; e.g. 1, 2, 3, 4, etc. For 15 spoken words, they might count up to 30. This was used so Fellini as director could get them to redo a scene with alternative physical actions. Example, "Do it as you did it before but this time at 25, turn and bow your head to the left."
When employing this film-making protocol, directors such as Fellini could keep up a running commentary; shouting direction to the actors on-the-fly since there was no reason for a 'quiet on the set' environment.
The post-production voice-over synching was noticeably lackadaisical in Italian films.
Composer Michel Chion has observed that Fellini particularly exploited the tendency of Italian films of the post-war period to allow considerable freedom in the synching of voices to lip movements, especially in contrast to Hollywood's perceived "obsessive fixation" with the matching of voices to mouths: "In Fellinian extremes, when all those post-synched voices float around bodies, we reach a point where voices--even if we continue to attribute them to the bodies they're assigned--begin to acquire a sort of autonomy, in a baroque and decentered fashion." In the Italian version of La Strada, there are even instances when a character is heard to speak while the actor's mouth is shut tight.
Wikipedia > La Strada > Sound
I'm not sure if the lack of synchronization reached the level of a B grade Kung-Fu or Godzilla movie from the 1960's but it certainly seems that synchronized voice-overs were not a high priority.