There’s no failsafe way to prevent your hands from swelling while walking, but there are some things you can do to promote blood flow and keep your hands warm, therefore stopping some of the swelling from happening in the first place. These include:

Making sure you don’t wear anything too right around your wrists or hands during exercise. See if you can comfortably wear your watch wristband looser and remove any jewellery from your hands or wrists. If you wear gloves during exercise, make sure that they fit you but aren’t restricting your hands or fingers.
Moving your arms around occasionally during your workout to encourage blood flow, eg making forwards and backwards circles.
Purposely moving and stretching your hands throughout your workout, including stretching your fingers apart, making a fist with each hand and raising them above heart level a few times.
Taking a bottle of electrolyte fluid out with you to ensure you’re replenishing some of the salt you lose through sweating.

“Why do my hands swell when I go for a long walk?” A GP explains