Nowadays most .mproj files are projects from a software called iMovie Movie Maker (not from Apple) and this is the software you have to use in order to export the video to standard format. The free version leaves watermark though.
It seems that .mproj files from Mine-imator also can be exported as AVI videos when the project is finished. This looks like to be the only viable option how to achieve mproj to avi conversion, considering no other software likely supports this type of .mproj files.
Updated: July 4, 2023