Most mortgages are portable, meaning you can transfer them from your current home to your new one when you move. This is something you will need to discuss with your lender and they will usually need to carry out a valuation of the new property before agreeing the move.
There is normally a fee for moving your mortgage to a new property, but, again, this is something to discuss with your lender.
However, when you are moving house, it is the ideal time to shop around and see if you can get a better deal on your mortgage. This might mean remortgaging with your current lender or moving to a new mortgage provider.
If you do choose to remortgage when you move home, you will usually have to choose between a fixed rate mortgage and a tracker rate mortgage.
A fixed rate mortgage will offer a set interest rate for an initial period (often 2 years) after which you will switch to a variable rate of interest.
A tracker rate mortgage will offer an interest rate set at an agreed interval above the Bank of England base rate (or an equivalent). Again, this lasts for an introductory period with 2 years being common.
Which option you choose will depend whether you prefer the certainty of a guaranteed rate or prefer the idea of a rate which could go down as well as up.