I don’t know of any heavily compressed format than maintains the alpha at a “small” file size.
List of Video Formats with Alpha
VP6 (FLV) supports Alpha but you’re not going to want to do post production work with it. Since web browsers have moved to direct HTML player support it’s not used much on the web theses days.
I’m not sure how big video PNG with alpha would be.
I’m not sure why you’d want to try anything not widely supported. Even a free DropBox or iCloud account can support up to 5GB. You can send the files one or two at a time.
Personally I’d use ProRes 4444