Hi, I'm just new to this site but have been walking for years with and without my kids. We started my daughter when she was 3 and my son was 9.
I lived in the Lake District at the time and we started with low level walks and I spent a lot of time with a small girl on my back. The trick is not to put them off it by making it feel forced.
So accept you'll probably take loads of breaks and it will take you longer than expected to do a distance you might walk in no time on your own.
The next problem we experienced was the lack of decent walking gear for little ones, especially boots. Brasher do kids goretex boots but finding them is the problem. Trespass were our best option for waterproofs as kids just seem to keep growing and they are a reasonable price.
My advice would be to keep them warm, dry and fed and they should keep going
Cold hands have caused a few tears so good gloves (again hard to find for little fingers) are a must for cold walks.
My daughters 8 now and loves walking and will happily run up snow covered fells to beat her brother to the top. They still live in the lakes and she's on her 54th wainwright.
Hope this helps.