Probably not really feasible, unless it's done separately for each distribution channel, as even if you stick to Linux, the binary distributed for each distribution can be subtly different, depending on which configuration options are selected.  Just musing, but I wonder if part of the compile process could include a script to check the "most recent" version of that variant, and if a newer version exists, (optionally) download and install it.  The script would have to be hand written for each compilation path (i.e., Win, Mac, Linux distribution, and stable or master) but the program could call that script on startup, if it exists.  Certainly worth some thought.