The foaming hand soap containers in question have a little disclaimer on them stating not to refill them with regular hand soap or they will clog. This is indeed true, however I've noticed a lot of people either throwing away the empty plastic container (what a waste!) or refilling it with regular hand soap, after which they just don't work.
The process is incredibly simple - the foaming pump works because it injects a bunch of air into a small amount of soap as it pumps. Most hand soap is too viscous to accept this air, and it just causes problems. Solution? Make a solution! Water down the soap!
Step 1: Choose a soap
Tip: Add the water slowly, so as to not agitate the soap too much - if it mixes while you're adding water, you will have a hard time getting it to fill without overflowing with suds everywhere.
Step 2: Agitate
Agitate slowly, or violently, however you prefer. Just mix until all the soap is 'dissolved'.
Step 3: You're finished!
Now you're on to using less soap, and wasting less plastic!
Behind the scenes: Soap itself doesn't really clean your hands. Soap merely makes most dirt more happy to mix with water than it is to stick to your hands. Hence why this product is so great. You can start with the foam (already 4:1 mixed with water) and do most of the hand cleaning with that, then simply use a LITTLE tap water to rinse off. Water savings too!
Downside - its more work. Alternately, you could prepare some watered-down hand-soap in a spare 4L jug, and keep that handy to refill your pumps.