I have never done this and had no idea (cute doggie BTW), but I did some research and here are some ideas I’ve found. Most of them would wear or at least wash away, but it sounded like the folks just retouched them when needed.
A lot of folks use makeup. Blush, eyeshadow being the ones I saw mentioned. They applied it with a brush or their finger.
Colored pencils were mentioned. One way was to wet the pencil a bit and work with it that way. Another idea was to use an emery board to rub your pencil on to produce colored “dust” that you could apply with a cotton ball.
One person mentioned what she called a pencil crayon, this is oil based. They might be had where art supplies are sold. Cray-pas is one brand, Artista another.
One person said she used fabric pens like they use on tee shirts. This might be more permanent, but I’d test it out on a swatch first. I suppose that would be good advice in general, to test any of it out first.