During the Adventure
Check your Expectations
Talk to any adult about why they snowshoe and you'll hear some basic themes: solitude, exercise, camaraderie, excitement of winter exploration, great way to get out during the winter, etc. When they first take kids snowshoeing many adults are excited that they can pair a winter workout with family time. Although this can be true with older kids, younger kids often don't have the attention span to "just" walk on snowshoes. Thankfully snowshoeing has loads of potential to turn a "boring" walk into an exciting adventure. Be patient and focus your expectations on having quality, fun family time rather than summiting a mountain. See Activities for ways to keep kids engaged.
Give Kids a Sense of Self-Sufficiency
At the trailhead, make sure that your kids know this snowshoe trip is an adventure and that they are now big enough to hike on their own (if they are the appropriate age) rather than be carried by you. You can also give them a sense of ownership by letting them carry their own pack with some water and/or their sack lunch. Communicate it all with a spirit of adventure and they'll get excited to participate.
Strap Snowshoes Right the First Time
Before you start out on the trail, do a thorough check of everyone's snowshoe bindings to make sure they are snugly attached.
Manage the Group
If you're snowshoeing with a group of kids, it is important to have at least two adults - one posted at the front of the group and one at the back. You might also encourage the kids to pair up with "adventure" buddies. Buddies make sure they stay within site of adults or can be in charge of checking noses, cheeks, and fingers for frostnip (the beginnings of frostbite).
Hydration & Snacks
You planned ahead and brought plenty of water, snacks, and warm beverages, now just remember to nibble and sip away! Normally thought of as a hot weather thirst quencher, water is actually an essential part of staying safe when you're snowshoeing. Kids might be having so much fun in the snow, they might not realize how thirsty their body has become. Snacks are the same way. Best to curb the thirst and hunger before crankiness and dehydration sets in. Take frequent breaks during the day and keep snacks and water handy. Consider getting kids their own water reservoir with tube for their backpack. Both adults and kids alike are more likely to drink more water throughout the day if it is easily accessible through a tube attached to a backpack strap. After each snack or lunch break, do a quick gear check to make sure all gloves, hats, jackets, etc. are accounted for before continuing on the trail.
With all those layers, it can be quite a chore to make a bathroom trip. As much as you can, plan for bathroom stops that will be easiest on the group (i.e. before leaving the house, before leaving the trailhead, and during "check-in, shake down" times). Just like with snack breaks don't forget to do a quick gear check. For other "bathroom" tips check out Leave No Trace.
There are lots of things to keep in mind to ensure a safe, fun snowshoeing adventure. When snowshoeing with kids make sure to check for cold noses, ears, cheeks and fingers often, that each kid has a whistle, knows when to use it, and knows the boundaries of play (i.e. stay on the trail or at least within sight), and that everyone stays properly hydrated and energized with water and snacks.