As Geoff has identified, there is both a medical and a wardrobe component to this question. I have a mild case of Primary Raynaud's, but it sounds like this runner is dealing with more severe symptoms. To clear my conscience before I address the wardrobe component, I will say that my understanding is that Primary Raynaud's is not dangerous and can be self-managed, whereas Secondary Raynaud's can be related to other underlying conditions, and should be evaluated by a doc. Assuming this reader is all set with their medical evaluations, here are my thoughts on gloves and layering;

It sounds like sweating is a primary issue here, as it so often is with thermoregulation. A pair of moisture-wicking gloves seems a good first step, whether a synthetic wicking material, or the classic warm-when-wet material: wool. Bringing along a second pair of gloves also strikes me as a possible solution: wear the heavy ones to start, and as soon as you start feeling too toasty, take them off and make the quick transfer to your medium-weight pair. In the winter, carrying an extra pair of gloves won't be too much of a pain, since jacket pockets will be readily available. Possibly even have a third pair nearby for post-run, so you can keep your hands in warm, dry gloves as your body cools off.

I've also found that core temp really matters for managing my cold extremities. I develop vasoconstriction most commonly after workouts, when my core has been super-heated by exercise, which I've now stopped, and my body hasn't yet figured out that it needs to start making its own heat again. That's when I'm really in danger of white and blue fingers. If you also find post-running to be a tricky time, I recommend excessive bundling of the core. So, off with all the wet clothes (sports bra included) and on with layer upon layer of wool and puffy things... pretty much as soon as your feet stop running.

I suspect cold skin anywhere on the body may signal the brain (or, specifically, the hypothalamus,) that we're cold! So perhaps if other parts of your body are cold, your brain is getting the message to shut down the blood flow to your hands, even if your hands aren't cold. Following this line of thinking, starting out plenty well layered, with wind-protection, may help halt the reaction before it begins. This is speculation. But if you don't already protect your limbs and face as well as your core, that could also be something to try. Face creams like Dermatone provide a buffer between the skin and the wind, and though they are slimy and unpleasant in texture, they do work well. I use them frequently when ski racing.

Good luck! As an aside, if you, reader, or anyone else with Raynaud's happens to be an xc skier as well as a runner, I would be interested to hear if you notice worse symptoms in one sport or the other. It's generally worse for me with running, and I have wondered if that has something to do with more dramatic arm movement/engagement in skiing, which keeps the blood flow greater and temp higher from the start. Follow up if you wish!