Hi Blueboy,
I moved my Mum with Alzheimer's from Scotland to a care home near me in Yorkshire in 2013, and one issue you might like to think about is the local authority funding allocation which is available for personal care in Scotland (where it is free) but not in England.
As in your case, my Mum was self-funding and so in theory she could have moved anywhere without the involvement of Social Services, but I was advised by the Scottish LA that if she did have an assessment (care needs and finance) by Social Services in Scotland before moving, the Scottish LA might still cover the costs of her personal care. I followed this advice, and although it involved filling in several forms and delayed the move by a few weeks, it was worth it as the Scottish LA agreed to pay £241 per week for Mum's personal care towards her care home fees in England.
Sadly my Mum died just over a year after the move, but the Scottish LA would have paid the personal care allowance indefinitely (and would also have taken over her full fees if her resources had run out). They liaised with the care home about her needs before she moved, and two social workers even came down to visit her after six weeks (it was the first cross-border placement the LA had made, and they were interested in seeing how the system worked).
My Mum was in a community hospital before moving, and she was transported down by a Scottish Ambulance Service minibus (without any charge to us). This was also arranged by the social worker who did the assessment. The reason that the LA was so helpful was that Mum had no close family left in Scotland, and the assessment showed that it was important for her emotional needs to be near to me. This might apply to your family too, if you are going to move south.
If I had not involved Social Services, I was told that I could just arrange everything individually for Mum, pay all of the care home fees and transport her there myself (or pay for a private ambulance). In that case, it would eventually have been the English LA which would have had to take over her funding.
I hope that all goes well for your Mum and that you work out the best way to arrange the move. The move south worked out well for my Mum, as she had a very happy final year when I saw her every day, so in my view you are doing the right thing.
Best wishes,
Blackcat