I've inherited several tasty family recipes that are casseroles with canned soup as a base. Is there an easy way to sub a more natural base?

I make a bechamel or a veloute to sub for that canned soup apcray. With a little practice, making them from scratch doesn't take much longer and tastes SO much better. ;)

For cream of chicken soup - I use 1 tablespoon butter
3 tablespoons all-purpose flour
1/2 cup chicken broth
1/2 cup heavy cream of half n half
1/4 cup milk
salt and pepper to taste
maybe add shallots or garlic to taste
Make a roux by melting butter, add flour, then add chicken broth and cream or half n half, and milk. The milk keeps it from separating.

In regards to cream of mushroom soup, I make the same roux only I saute mushrooms in butter before adding the flour and liquids.
drbabs gave you a great starter for a cream sauce, also.

I use home-made soups over rice often as a way to use a left-over in a way that it becomes something new. Add any kind of cheese that goes well, and there you are.

You could certainly make a flavorful cream sauce and use that as the base instead of the cream of whatever soup. Here's an example of a recipe that does just that:
http://www.food52.com/recipes/9151_tuna_noodle_casserole_redux