I live in an 8x18 tiny house on wheels with my husband and we are expecting our first child. We've lived here for 6 months.
I do not recommend pursuing a tiny house unless you have somewhere to park it. And you should be prepared to pay rent for the land on which you park it. Tiny house parking seems to run from 250-600 per month depending om location. Monthly campground rental seems to run around the same. Used travel trailers are usually much more affordable. I would consider a used travel trailer and rehabbing the interior for the most cost effective option.
We chose a THOW because we already had a place to put it and because we live in a cold climate and four season travel trailers are really expensive.
Anyways, it is definitely possible to have a tiny house on wheels with a child! Check out www.minimotives.com. She has a toddler and one on the way. While I a, confident that we could accommodate a single baby in our size house, I don't think we would be able to have an older child or multiple children comfortably, especially considering that we live in a cold climate.
One option that we have is to purchase another tiny house and connect the two with a screened in porch. Andrew Morrison has a similar set up for his family of four, including two teenagers. I would place the two houses in a long line, and add an exterior outhouse off of the screened porch as well. Our current tiny house would be converted into a sleeping quarters, with our bed in the loft (as it is now) and the entire downstairs converted into a bedroom/playroom for the kids. The second tiny house would be larger (8x24) and contain a kitchen, dining room, and living room, as well as two lofts. When the kids get older, we will move our bedroom into the bigger tiny house and the oldest kiddo would have the loft as his bedroom.
That's about two years off, so hopefully we will have time to build up some savings to make it happen!