My Lincoln stick welder sits on caster wheels attached to the original wood packing on its bottom. It doesn't leave the garage. If you attach it to a hand cart, there shouldn't be any stability issues, and you can weld some gizmos on the hand cart to hold other stuff.
I'm sure there are numerous examples of welding carts on the Miller welds and the Hobart welders websites.
The mig welder cart is a little different critter because you need to protect your mig hose from damage and eventually you'll get a welding gas bottle that needs to be moved with the welder.
Your cart design should be based on how you personally use your welder--whether it goes outdoors and needs big tires, what stuff you need to weld (grinder/etc), where you keep your rod, helmet and gloves, etc.