Snowshoe running is fun because it's hard. You're sweaty and exhausted in a small fraction of the time it takes to wreck you on a "normal" trail run. Because you get so hot so fast but you're out there in winter weather, choosing the right clothes for the sport and the conditions will not only keep you comfortable, it will keep you safe.
Feet: you're going to get your feet wet--managing moisture will keep you warm, even stave off frostbite in the worst conditions. I use a GoreTex runner with my snowshoes. Less moisture gets in, but my sweat has an opportunity to wick out. Because GoreTex shoes are warmer than others, I wear a surprisingly thin sock. This keeps me from getting too sweaty in the shoe and chilling from the damp. I will also put a Dirty Girl Gaiter on to keep the snow off my socks. Snow will pill and cling to exposed socks and even though these gaiters are not waterproof, they will shed the snow and keep your feet dry. Plus, the are light and flexible enough to wear under your snowshoe bindings.
Bottoms: This is the garment with the most room for error. Again, snowshoe running will make you incredibly warm incredibly fast, so you need to dress light enough to shed excess heat and stay moderately dry. What will complicate this are the snow conditions. The tails of your snowshoes will kick up the snow and the more powdery the snow, the more you'll have stuck to your backside. I've had success with windproof Nordic ski tights (which shed snow better than running tights), uninsulated soft-shell pants, and on days when the snow is sticky and therefore less likely to powder-coat my legs, running tights with maybe (usually not) a pair of shorts over them.
Top: You'll be tempted to dress as warmly as for other winter runs, but remember that snowshoe running will turn up the heat. I keep my warmest layers at the trailhead, or carry them if I'm managing my safety in the backcountry. I will typically wear a long-sleeve shirt with maybe a light wind-blocking layer over it. Full and quarter-zips are pretty magical for venting your chest. As spring approaches and the temperatures really warm, I like to wear a tee and start with arm sleeves which I'll remove after the first mile or two once I heat up.
Hat and Gloves: Again, I tend to wear a little less than what the temps suggest and carry a heavier layer in my pack if I am managing my wilderness safety. The only special thing to consider is that falling can leave you with very wet hands. Choose a glove with some waterproofing or that will hold its heat, like wool will, when it's wet.
What does this look like in actual practice? The 2019 Moose Dash 5k was held in fresh, powdery snow, on groomed trails, on a day that the thermometer maybe reached 20F. Normally, I'd carry some layers for an emergency in these temperatures, but the inherent safety of a race course meant I could travel light. I kept on a puffy down coat until moments before the gun, shedding it at the start. I wore a long-sleeve tee, tights, a buff, gloves and gaiters. I intended to shed my running skirt too, but spaced it. In the end, I was glad to have kept it: I kicked up more snow than I expected to on a groomed trail, and the skirt kept my backside warm while my top half vented heat. And vented a lot of heat. I would have been underdressed for more casual or longer run, but I ran this 5k uncharacteristically hard and was thankful to be dressed so light that I could keep pushing without cooking.
Ultimately, you're going to experience a lot of trial and error. As you figure out how to dress for snowshoe running, you'll have days when you are wildly under- or over-dressed. Just keep your safety in mind as you are learning, and carry extra layers and an ultra-vest so you can carry layers and add or shed clothes at any time. Remember, once you're wind-chilled or sweat-soaked you're vulnerable to hypothermia, so make adjustments as soon as you feel uncomfortable.