I've seen many-a-person annoyed or otherwise dissatisfied by the fact that your posts on one Fedi instance don't move with you when you switch instances. (There's some obvious technical reasons for why this is the case that I'm not going to explain here.) This is why having your own website is so important and can be so valuable! If there's something you've posted on social media (or wherever on the web) that's worth saving, then why not make a copy of it and have it published on a web site? Your website!
I like to refer to this as "Reverse Syndication", but for those of you that speak #indieweb, you've probably heard of #PESOS ("Publish Elsewhere, Syndicate Own Site"). Basically it means that what you publish FIRST on social media, you make a copy of on your site. This could be as an archive, a canonical representation, a data ownership/soverignty play, w/e!
I've used a ton of social media sites, sites like Reddit, and switched Fedi instances multiple times. Across these years (at least since I've been bathed in the glorious light of the Indieweb), I've been PESOS-ing things I think I want to hold on to to my site. The vast majority of these things I file away as "notes" (https://shellsharks.com/notebook). This allows me to A. archive things as I've discussed, while B. not "junking" up my main RSS feed, or blog post timeline with silly reverse-syndicated social media posts/micro-blogs. It takes a little bit of effort, (unless you have some fancy-pants automated PESOS-ing system), but the pay-off is huge imo.
So let this be your answer for "I wish I could transfer my old social media posts to my new social media platform"
Another thing you can do, specifically with Mastodon (but likely with other post archives), is turn the entire archive into something you can publish on the web. Look at this! https://shellsharks.com/toots/infosec-exchange/shellsharks/. This is all my posts from my infosec.exchange days in one gargantuan web page on my site. In all its insignificant splendor. (More details on how I did this are on that page).
Happy data owning everyone!