In most cases they will use the same flight number all year long. Not sure if it is the exact same plane every time (I would doubt it), The Delta, United, American and Air Canada flight numbers will stay the same. The only ones that could change are the Atlas Air or Paradigm Air. Some times they change, some times they don't.
planes (hardware) change, flight numbers stay the same from the planespotting I've done. I suppose if a team has a custom livery plane, the hardware may even stay the same as well but for the two teams I planespot, they don't have a customized plane for their use so the hardware changes that carries out the flight number.
What you will notice is sometimes during a long homestand, you might see that flight number used for another client but then used for the MLB team again when the homestand ends and they need a flight to their first road city of the new roadtrip.
Track the hardware sometime - it's cool to see a MLB flight take a team somewhere, then that hardware will change flight number and go to commercial paying customer service so you don't have the plane sitting idle at the city of the MLB team for days or even a week or so. When the MLB team is ready to move to the next city, the flight number they use will return and it'll be different hardware in the cases I monitor.