After hours of searching, I finally figured out that the free version of Any Video Converter will convert video files I have into ones I can view on my Walkman device.
I have used the mpeg4 option at a resolution of 320×240. I chose a bit rate of 112 and a frame rate of 15.
Using that method, I get a pretty good picture. A roughly 700 MB .avi file became a roughly 64 MB .mp4 file. Puzzlingly, a roughly 400 MB .avi file became a roughly 120 MB .mp4.
I’m afraid I don’t care enough to figure out what caused these differences. To my knowledge, I encoded these files using the same method.
Anyway, I hope someone finds some of this useful!