Could a slave or indentured servant become free by running away or moving to another state?

Yes, if they were successful in moving/running away to a non-slave state, they would be free.

It depended on the state. Some states yes. Once you were there you were free, and your owner couldn't come and get you. In other states however, if your owner found and caught you, you had to go back. It's different for indentured servants though. They servants because they made a deal with their owner "You do this for me and I'll be your slave for so many years" They couldn't attain freedom fro running away.