Why would an MP3 sound file drop in bit rate when transferred via Dropbox?

That sure sounds like it could be the problem. I’ll keep this in mind for next time. Thanks!

Yes, that is 99% likely the problem. This is an issue I deal with regularly as an audio engineer. Depending upon the encoder your engineer is using, the default is most likely VBR, and s/he either needs to de-select VBR (Pro Tools) or actively choose CBR (Wavelab/Myriad/etc).