This is based on my own experience, the quick answer is, call AEA and ask.
Moving a show from LORT to Broadway is complicated, because it's changing producers, changing contracts and there is often a down time that seems to make people forget there was a previous production.
Unless you have written in your contract, it is best to assume that you are not guaranteed a seat in the next production. There is nothing outlined in the LORT contract book about it, but there is precedent setting situations they maybe referred to. But every situation is different. (They are clear guidelines about inter-LORT transfers, but in reality nothing spelled out clearly for transferring a show form a LORT contract to another AEA Contract.)
I always look at it as, if I am doing a LORT production, I have a contract with that theater. If an outside commercial producer then takes the show, I would have to enter a new contract with them. Often, a commercial producer may come in and clean house, but in reality, if you know the show, then why wouldn't they keep you? (Unless there other reasons). So, even if I had a contract with, let's say The Alley theater saying that I am guaranteed a spot on the transfer or a buy out, unless The Alley itself is producing the Broadway production (unlikely), then I am not sure legally where you would stand. At the very least, the very minor amount of the buyout could happen.
Every LORT to Broadway transfer I know about has involved a second set of negotiations, but all the ways I worked on were very carefully constructed to protect the commercial producers' interests.
In short, it's not automatic, nothing rarely is.