when you send the ecash, in current spec, the mint url is embedded in the note.

The ecash you create from mint A is only 'redeemable' from mint A. You and the final recipient need to communicate with mint A (to mint/melt)... mints B, C, D won't care for ecash from mint A, and vice-versa.

When you share ecash with another user, if they  dont have that mint connected they will be prompted to add that mint to their list (asking whether they trust them or not).

If the recipient wants to move ecash from mint A to mint B (as they might trust B more), then they would do a swap between mints. Under the hood a LN invoice is generated and paid between the mints, sats will be sent, and new ecash minted to user in mint B. fuckin' magic.