As you are moving to a new district where there is an 11+ – if this is an 11+ region (i.e. one test for ALL the grammar schools in that region) then you will need to register with the LEA and apply to take that test.
If there are many independent grammar schools in the region that you are moving to or the LEA grammar schools have their own different entrance tests – then you will need to apply to each school separately. Perhaps you can find this out by phoning the LEA of the new region.
The 11+/entrance exams are always taken in the grammar school or district that they are applicable to. I have never heard of a test being posted to anyone outside the 11+ region or a child being able to take a test outside that region for a school in a totally different region.
Please do be careful not to miss the deadlines for your application(s)- look them up online or ask the LEA/school(s).
Which region/district are you moving to?
Hope that helps you.