How to move an account to another XMPP (Jabber) server?

I have an XMPP (Jabber) account on a public server that is not reliably reachable any more (looks like the server is not really maintained any more). Therefore I want to migrate to another server. I already have registered an account on another server; but how do I move my contacts there?

Maybe this is a "pointless" question; but my understanding is that other people know that I contact them via the account on the old server; and if I use the new account, wouldn't they need to accept the contact request first? Also, how do I tell my XMPP app to contact them via the new account at all?

I'm using the Conversation app under Android.

Maybe this is a "pointless" question; but my understanding is that other people know that I contact them via the account on the old server; and if I use the new account, wouldn't they need to accept the contact request first?

They would, yes. XMPP does not have any hidden long-term user identity that would let others associate your previous address with the new one – other than the user@domain itself. So if your new account is at a different domain, then it's a new identity.

(Yes, there's OMEMO, but that's built as a separate layer on top of XMPP and doesn't interact with the underlying system of accounts and subscriptions; it just exchanges generic "events".)

In that regard it's no different from email, or even ActivityPub "fediverse" (the Mastodon/Misskey/etc. network, which does have cryptographically bound identities and a half-baked migration system, but migrating to a new account still involves requesting new 'follows' in the outbound direction).

Also, how do I tell my XMPP app to contact them via the new account at all?

Depends entirely on the XMPP app. It ought to have a feature to export the "roster" or "buddy list" or similar, and to import them back (to the new server).

Since Conversations doesn't have that feature, set up both accounts on a different client such as Gajim which has "Preferences > [account] > Contact List > Export". The roster is saved server-side so it should automatically show up in Conversations once imported.

But if the old server is really not reachable anymore, then I'd try to find where the existing client caches its contacts (most of them have a local cache) – with Conversations you might need to use its "Backup" feature, although I'm not sure what kind of compressed/encrypted output format it uses.

In the end, since there's no true "migration", manually typing all of the contact addresses into e.g. Gajim will achieve the same thing as export/import.