You didn't mention what kind of lease or rental agreement they have/had with the previous owner. I used to acquire units all the time that had tenants with no written agreements (tenants at will). If they are tenants at will, you are (somewhat) in luck. I say somewhat because any move that might involve eviction can be quite unlucky for you indeed, if you are in a tenant paradise state like MA (or, unfortunately, in your case RI).
In practically all these cases, the first thing I did was serve them with a 30-day notice to quit to terminate the old tenancy. For the ones I wanted to keep, I would offer to create a new tenancy, which would be a written tenancy at will (no leases, thanks), and usually at a higher rent, since their rents were all usually below market. Also, I would put the requirement in there for a deposit (with signed statement of condition) and last month's rent. Often, the landlord had failed to collect these or had already used these monies for occasions where the tenant had not paid (improperly, of course, in the case of a deposit). For the ones I didn't want to keep, the 30-day notice got the eviction process started.
For multi-family rentals, keep in mind that often the reason a landlord is selling out (sometimes at a ridiculously low price) is exactly because one or more tenants are creating problems, sometimes very serious and costly ones. It's always a good idea to do a little research (no, a LOT of research) before you close on the property to get a sense of the hornet's nest you may be buying into. If you think it might be so, see if the landlord is willing or able to remove the problem tenant(s) before closing. If not, at the very least have a talk with a local attorney that specializes in evictions to get some sense of what you are up against and what the worst-case scenario might turn out to be for you cost-wise and aggravation-wise.
Just a few thoughts from an older but wiser ex-landlord.
