> When you move because you don't like the instance anymore, how do all of your connections follow you?

You use the built-in[0] migration feature. Create an account on your new instance, point it at your old one. Then point your old account to your new one.

[0] Built in to Mastodon, anyway. YMMV with other Fediverse software.

And if the instance-administrator on the old account decides they don't like you and drops the account or points it to /dev/null?

That risk exists at everything you do not host yourself on the internet.

The risk is very small for email when you own your own domain. One can switch "instances" at will with just a DNS update.

You can do this with Mastodon too. Point a domain at one of many managed mastodon services or host your own.