what a daily walk you have!
I could be wrong (often difficult birds, and your photos are not brilliant), but on the basis of the second shot (same bird?) I would say that this is a female yarrellii (Pied Wagtail) not alba (White Wagtail).
Its good exercise and the dog loves it too. He spent 1.5 hours in the sea being teased by the sea gulls. He ignores everything else. The sea gulls seem to appear from no-where. They seem to recognise hime and start the big tease.
This leaves me to walk down towards the sea when the tide is out and observe what is around. The birds seem to be getting less nervous now and I can get much closer. This is just as well as I am pushing a cheap kit lens to the limit. Its not too good on detail unless you fill the frame! I have a 70-300 lens but it was faulty from when I received it. I hope the replacement is in stock by next week. Its soul destroying to make opportunities but not be able to realise them because of the lens. At least its good practise for when I get better quality optics.
Apologies for the photos. See my reply to Joe.
I spent ages debating which type they were. I chose the white because of the grey back but now, on reflection, you may well be correct. When you have to make a choice on your own about such similar birds, you can mislead yourself. When I get my new lens I shall try to find them again and get better images.
These are not easy birds, with different ages, sexes and subspecies it can be VERY difficult to be precise. Just take a look at "Yellow" Wagtails!!!
Yes, this bird is a female Pied.
Don't forget that female White lacks the sharp bordered cap and nape and is actually very similar to your female Pied, just slightly paler grey and a cleaner black into grey on the head
Just in case...