Usually this is because of the source video being variable framerate which got played with constant frame rate in Premiere, desyncing audio and video, because the audio doesnt have a variable framerate.
You might try to convert the video with Handbrake to a constant framerate first.
Said this, I have the same problem, but with constant framerate.
IMHO... any format that is temporally compressed is going to be vulnerable to these types of issues. H.264/mp4/mp3 audio included. I know that Adobe says you can cut with these formats but I avoid them like the plague and transcode to a format that has full frame information like ProRes or Uncompressed 10bit first.
Converting to Prores on a W-machine is not an option.
uncompressed 10bit for webcam footage?
ProRes actually is an option with the right third party codec... but that is aside from the point, a non-temporal compressed codec will play better with any NLE than temporal compressed footage. Pick a codec that works for you. 8-bit in this case would be fine.