Using a Verizon Phone on T-Mobile - Who Updates the Firmware?

[toolworker]

I've got a Galaxy S4 from Verizon. I used it on a T-Mobile account that I got for foreign travel and it works fine.

Now suppose I switch to T-Mobile permanently, and a new Android version comes out.

Can I put the Verizon SIM card back in and get updated, even if there's no Verizon account for the phone?

Or is there another way to get updated?

Or am I stuck with the current version unless I go back to Verizon?