I don't really use Mastodon, but I jumped on a couple instances to check the process of finding the post on a different instance. The posts in mastodon again have a different URLs to the same post on different instances just like Lemmy does so I just I wanted to figure out how they did it. It was simple and intuitive.
e.g. I may be browsing a post on mastodon.social while logged in to mstdn.social account and want to switch to an account on mastodon.world before replying.
in (https://mstdn.social/@wonderofscience@mastodon.social/110678720030292635)
click the elipsis ...
Copy link to post
This copies the link to the original post in its original instance (https://mastodon.social/@wonderofscience/110678719111063776)
Now I can go to mastodon.world and search the link we just copied. This gives one result which contains the post: (https://mastodon.world/@wonderofscience@mastodon.social/110678720000699504)
*Note: it seems you have to be logged in to the Mastodon instance for the search to work.
*
Now you should be able to reply from that instance. I suppose these are the steps that the Fedilab takes for switching accounts.
I was scratching my head why searching this post from lemm.ee (https://lemmy.ml/post/1769528) wasn't working for me. Then it hit me, the post original instance is based on the poster, not the community. There is even a Fediverse star that gives you the link to the original post which is actually (https://reddthat.com/post/326735).
You can actually search (https://reddthat.com/post/326735) in the instance that you want to post to and it will be found using the search bar. It's just not as fast as Mastodon. I have convinced myself that this isn't as big as a limitation as I thought, it just isn't as trivial as a simple URL change.