If I bring my own phone that is unlocked to a carrier, will they lock it with them and do they have every right to do that?

I have an iPhone that I just had unlocked so I can go switch to another carrier, but I'm wondering what will happen when I take it to another carreir to put service on it. Will they lock it with them?

No, the carrier you go to can't 'lock' your phone. Generally speaking, from what I've read, once your phone is unlocked, it can be locked again. You'll get (or buy) a SIM card from the other carrier and open an account with them. Once you put in the new SIM, the phone should accept it and it should start working.

Did you "jailbreak" unlock it or unlock it through your previous carrier? The new carrier shouldn't wipe out the "jailbreak" but the best way to prevent it is to have them hand you the sim card and you install it yourself. If you hand it to them, there's no telling what they'll do.

The will not try to lock your phone.

Ask the carriers direct. It is likely that each carrier has a different policy about locking.