I can understand how jerkiness/flicker could be introduced (differences in the frame rate of the original film, the frame rate at which it was captured, the frame rate of your PC monitor etc.). But don't know why some captures should exhibit A/V sync errors and others not...
Is the issue only seen in Studio - if you play one of the "problem" recordings in a media player does it stay in sync? If it stays in sync in media players, you could try processing it using HandBrake (maybe there were dropped frames in the recording so it appears to be a variable frame rate)?
Changing video bitrate for the export would not have any effect on the amount of flicker or out-of-sync; higher video bitrates could improve picture quality by not throwing away as much fine detail or movement that was present in the video.
If you've got newer versions of Studio on that PC too, would MultiCam Capture Lite do a better job of the screen recording than Live Screen Capturing?
Does the website offer a way of downloading the films- or could you use a third-party video downloader app to grab them?
Regards,
Richard