Many energy companies will let you take your fixed tariff with you if you move home, but your prices may change if you move to a different region, so check this with your supplier before you move.
In some cases, you won't be able to stay on the same contract when you move, for example, if you move from a property with a smart credit meter to a property with a pay as you go meter, your fixed tariff may not be compatible.
If you move home and choose not to take your existing tariff with you, most companies will charge you an early exit fee if you have more than 49 days left on your existing contract. If you can't take your tariff with you, you usually won't be charged the exit fee, but it does vary by supplier, so check with yours.