I bought these for just that purpose (but also to run with the dog). Been great in the rain (not tested in anything torrential yet) and early mornings kicking through the wet grass.
I go with Asics trail running Gore-Tex, very grippy sole with decent waterproofing. Usually come in black or bright colourways. Often in sale on running websites
I need some water proof (resistant) footwear for the winter dog walks.
More long grass and muddy fields, than fording rivers.
Currently drawn to Vans MTE, which also have Gore-Tex membrane, so should be more waterproof than others?
An recommendations for wintery trainers that will keep my toes dry, and me upright chasing a puppy across a muddy park?