Putting smaller wheels on a larger frame
Putting small wheels on a bike designed for larger wheels will provide you with an array of design opportunities.

Here's a rough illustration of the objective


Let's go through some of the issues. (not intended to be an exhaustive list)


Brakes
Pedal ground clearance
Chain ring ground clearance
Gearing


Brakes
If the bike has rim brakes they would become inoperable with small wheels.  

Options
- Go with coaster brake 20 inch wheels and only have rear brakes.
- Do some creative wielding and install brake mount points that would allow calipers / v brakes etc. to function.
- Rig disk brake mounts on the frame and build disk brake 20 inch wheels.

If the bike has disk brakes the best option would be to build disk brake 20 inch wheels.

Pedal ground clearance
Smaller wheels put the pedals closer to the ground. You'd need to find a crank that is short enough to provide the correct clearance. If the pedals only hit on while turning you could learn to pedal only while going straight and keep the crank parallel to the ground while turning.

Chain ring ground clearance
Depending on what you start with this may not be a problem. If you find that your chain ring hits you'd have to go with something smaller.

Another way to address ground clearance is with some creative frame modification. If you cut the frame so that you can move the bottom bracket higher and then wield it all back together you can keep your current configuration.  

Gearing
Smaller wheels have the effect of making all your gears lower and reducing your top speed. This may not be a problem for your situation but it's something to keep in mind.

A project like this is a great way to learn about the interaction of design choices.