"Push to relays" does publish a kind 30002 event to public relays, so in theory, you should be able to pull it on other devices. There might be a bug, haha. This process is too cumbersome, so it will be removed in the future. It should sync automatically.
After poking around a bit, I did discover that if you already have some kind 30002 relay sets created, Jumble can import them. When you are on the screen for editing your relay sets in Jumble, you just have to select "Pull from relays" at the top right, and it will show you the relay sets you have saved.
One would expect that the "Push to relays" option would do the opposite, taking a relay set you have in Jumble and saving it as a kind 30002 relay set, but it doesn't appear to do that. At least, it was not available when I selected "Pull from relays" after Jumble had confirmed the set was pushed to my relays. Perhaps nostr:npub1syjmjy0dp62dhccq3g97fr87tngvpvzey08llyt6ul58m2zqpzps9wf6wl can shed some light on how that works.
Creating relay sets in Jumble on one device that can easily be imported on another would be very helpful. For now it seems you have to create the kind 30002 set in something like listr.lol first, and then you can import it into Jumble on any device.
Discussion
I like that you have a few defaults set up. I hope there is a way to keep those when you update to syncing a users' kind 30002s, even if it is just for users who don't have any kind 30002s already.
I plan to use a new kind 10012 event to store favorited relays and pointers to relay sets. I'll try to upgrade without affecting existing data. https://github.com/nostr-protocol/nips/pull/1848