Babies are generally pretty happy if they're right up next to you and moving, so dancing can work very well. We had good success with stretchy wraps (ex: Boba) and soft structured front-carriers (ex: an Ergobaby when worn on the front). I don't have any contra pictures, probably because if one of us was wearing Lily we were probably dancing together not taking pictures, but here's Julia wearing Anna in a wrap for international dancing at NEFFA:

Wearing a baby, especially a small one, we would dance a bit differently. Moving slower from the extra weight, but also more carefully and deliberately because of the increased vulnerability. We'd keep our arms a bit higher than usual to reduce the risk that a stray elbow might make it through to the baby. Other people aren't necessarily going to notice the baby, and space that would normally be vacant can need a bit of guarding.

This also gets into why, unlike front-carries, back-carries aren't a good fit: the baby is less protected. For example, if you're swinging (spinning with another dancer) a baby in a front-carry is moving slowly and is inside, while one in a back-carry is moving very quickly on the outside. And frame backpack carriers really don't work: they're not designed for this kind of spinning and babies can fall out. You're also likely less aware of where someone on your back is, since you can't see them or how they relate to the dancers around you.

Once kids get past the stage at which they're happy being in a carrier indefinitely (assuming yours like the carrier) they're probably not going to be up for a whole evening of dancing. We would often go just to the beginning of a dance, get in a few dances together and say hi to friends, and then leave at or before the break. Keeping toddlers out of the line requires a lot of attention: dancers aren't (and shouldn't need to be) looking down for them, so until kids know how to stay out of the lines it can be stressful.

As the kids got older I found myself carrying them in my arms more often. This does mean you have fewer arms available for dancing, but in my experience people are happy to accommodate. Being able to quickly switch between carrying them and not is very helpful, especially if they want to walk for part of the dance.

How workable this is depends a lot on your upper body strength: Julia was much less comfortable with this because she doesn't feel like she can do it safely with one arm. With an older child that can use their legs to help hold on, this starts to be a bit more similar to using a carrier and relies less on arm strength, though by that point they may be heavy enough that you don't want to dance holding them at all. My kids tend to be pretty light, and if yours are less so you might want a carrier to take the weight off your arms. Here's my sister carrying her youngest at NEFFA this year:

For most kids there's an age at which they are either too heavy to carry or aren't interested, but aren't yet old enough to handle the regular dances. Family dances, events with dances chosen to be a better fit for younger kids, can work well if there's one near you:

Sometime between 4y and 10y most kids get to where they can participate in the line at a regular contra dance. This depends a lot on how excited they are about the dancing, how well they follow directions, how familiar they are with the environment, and how well you're going to be able to help them through the dance. Lily did her first dancing around 4y but then decided she didn't like holding hands with strangers and only got into it again when dancing restarted.

See also: Miriam Axel-Lute's advice on how to dance with kids at contras once they're in the line.