Not sure what you are asking here. Do you want to manage the user script in a single place and have it sync on multiple browsers? Greasemonkey seems to support that here.

This answer also shows how to trick Greasemonkey to work with the same shared location regardless of browser/machine.

Thanks Alexei, this is not what I need --- I have updated the question to clarify.

OK. Does locally including (executing) a remote JS work for you? Check this

deleted user If you managed to make it work, please include an answer containing the solution, so that future users benefit from it. Thanks.

Alexei‭x I don't believe that @require USM function is what I need because using it caused an infinite loop in google.com in Tampermonkey for some reason and it might raise compatibility issues in different USMs. Please consider to delete this thread because the main concern of improving the question has been addressed.

If I understood correctly, require seems to be the way to go. If you're getting an infinite loop, the problem probably isn't require itself, but something that the required script is doing

hkotsubo‭ Even when the required script had only alert("hi"); the infinite loop behavior continued (until after about one hour), perhaps it's a caching matter; if required initiates cache then that's just another reason why I shouldn't use it.