We are walking in the Avon Walks for Breast Cancer around the country. Everytime we do our training walks (5 miles to 12 miles averaging 16-18 minute miles) our hands swell up a lot. Is there anything we can do to prevent this from happening especially during the Avon 39.3 mile walks?

Not really, short of keeping your hands high, like power walkers tend to. the issue is simply fluid building up rapidly in the hands becuse of the swinging motion (centrifical force) not allowing the fluid to drain back up as easily.
when you stand, there's no extra force, so the body can keep fluid moving upwards and fight gravity. but with the constant force of swinging the arms, they won't drain unless you help them.

Malacore hit it right on the head... I get this sometimes when I run.. I find I am holding my hands down by my belt... and I feel my fingers all swelled up...

when it happens I pull them up shoulder level for a while... (kinda look like a boxer running lol... )

Good luck...