Audio out of sync in Final Cut Pro
What do the original files show up as in a media app such as VLC? Does this info match what Final Cut says about the ProRes copy? The video frame rate, audio sample rate, and other information should match exactly.  Then, your FCP sequence settings should be identical. 

Another way to address small sync drifts is to razor-cut the audio and manually re-establish sync a few times on the project.