First up...DO NOT attempt to dry the shoes using any heat source (hairdryer, sunshine, oven, heater, microwave - yep...I've seen it all!) as this will cause the leather to dry out and could shrink the shoes (told you I've seen it all!)
Wipe down the shoes as much as possible removing any mud from the uppers.
Remove the innersole and stuff the inside of the shoes with newspaper/papertowel. Don't overstuff the shoes as this could potentially stretch the leather permanently. Paper will help to draw the moisture from the leather. Keep an eye on them and change the stuffing over a couple of times to get them dry.
I would suggest you leave them in the laundry to dry. Somewhere out of the way, but not somewhere that gets hot - they need to dry naturally.
Once dry, give them a polish with old fashioned boot polish with a brush (Kiwi brand or similar). Allow the polish to soak into the leather for a couple of hours before buffing off. Repeat the polish a couple of times so the leather gets soft and moisturised again.
If you have textile or canvas shoes, follow the same rules as above - however they won't need polishing :-)