I'm not a lawer. This is not legal advice.
-Meet the tenants and talk to them in person. Tell them you'd like to put them on your lease.
-If they say YES, then they sign and they're under your rules
-If they say NO, you have to honor the rules of the current lease. Reasons for no usually fall under time frame. Tenants typically want the flexibility and they'll look forward to the end of a lease.
If the lease is in place either way, just contact the tenants and let them know how you want to handle payments or what the new address is.
When it comes to the deposits, this can be tricky. Most owners won't want to come off the cash so they'll try to credit it to you at closing. This won't help you much since you'll still have to come up with the money. On our second deal , we communicated with the seller that we wanted a check at closing. On the closing day at the title company, we had to sign all the paperwork then the title agent left the room and the seller gave us a check.
Not a lawyer. Not legal advice.