To take footage directly from Avid to FCP take a look at Automatic Duck.
https://www.automaticduck.com/products/pifcp/
It’s designed to do this and runs about $500. Otherwise, you can export each clip individually by selecting every clip in a bin and choose File–>Export. Choose Quicktime, go to your options and set the Codec to Animation for best quality. Choose a folder to output to then be prepared for a really long wait.
If you’ve cut a sequence, simply mark In and Outs on the areas of the sequence you want to export, select the tracks you want to export, and do the same as before–File–>Export, choose Quicktime, Use Marks, Use Enabled Tracks, select the right field order, Animation Codec, select the proper size, Video and Audio (if you have video and audio), then select a folder and hit OK. Take a nap. Knit a sweater. Read War and Peace, because this will also take awhile, depending on the length of the sequence.
Warning: You won’t have timecode on any exported clips this way. Doing an UpRez or any type of batch capture in your FCP system will be impossible via the Quicktime export method. To be honest, it would be a lot faster and easier just to Capture and Cut on FCP or Avid instead of trying to port between the two. Unless you buy Automatic Duck. It should help (I’ve never used it, but it gets really good reviews everywhere).
Good luck!
Mike.