I used very basic prompts including words like "cute" and "happy" and environment descriptions but not much beyond that (no art style or specific colors of the characters). I felt overall the character consistency has room for improvement but could have been a lot worse. Midjourney's default "style" has a sort of specific flavor to it.

One of the downsides is how the characters look so clearly different in each picture. I don't know if there's any way around that, and might be that kids don't care about such things anyway.

I think that can be done by passing in an image as reference in the prompt.

Yes, you can pass an image (or more) and Midjourney will use it in the picture. You are also able to control the level of "creativity" Midjourney has when creating the image.