The short answer is yes, many mortgages are 'portable', which means you may be able to transfer your current mortgage to a new property. Although, in effect you will have to reapply for that deal, so if your circumstances have changed at all, then you may no longer qualify and you will be subject to a new credit score.
There are lots of costs to keep in mind when moving home, more so than for a first-time buyer: Estate agency fees, property survey fees, mortgage arrangement fees, solicitor’s fees, removal fees and mortgage broker fees.
When you move, your legal representative will pay off your existing mortgage in full from the proceeds of the house sale. You will need to start a new mortgage if you are purchasing a new property or “port” your current one to your next property if your lender permits this.
You can pay your mortgage off in full at any time, as long as you also pay any early repayment charges that may apply. Most lenders will allow annual overpayments of 10% of your mortgage balance per year, if you are still in your introductory, fixed or discount period. If you are on a standard variable rate you can usually overpay as much as you want.