It's simple. There are two kinds of wash cycles. Low temp and High temp. Low temp requires a 140F and expensive chemicals for the sanitation cycle. They also do not dry well. Most commercial machines will wash under 3 minuets cycle. The High temp machines have to have 180 or above for the sanitation cycle. That means they have a water booster in the machine. But house water should still be above 140F. If lower they use a bigger booster.
Now when you turn on the machine, water goes in through the rinse arms to fill the wash tank. Most have a sensor in the tank to check the ph to inject cleaning agent in the proper amount. When the wash cycle is triggered, whether a conveyor or batch machine. The tank has to be to temp and full for wash. If low on cleaning agent, the injection system will add cleaner. Different brands and models use anywhere from 60 second wash to 3 min wash unless the operator puts it into continuous wash for heavy cleaning. With door machines, At the end of wash, The pump stops and a solenoid opens the rinse valve. It also turns on the rinse agent pump. Remember that the rinse water dilutes the wash water. It also overflows the tank to carry off top scum. That means that it also has to add more detergent in the next cycle. With conveyor machines the rinse goes to the last wash tank, from the last wash tank to the next etc.
If you look in you machine, You will notice that there are two sets of arms. One is lighter with small nozzles for the rinse. The wash arms will be bigger and have larger ports to handle the water from the pump. Most machines will have a pump from 3 to 7 hp. I have run into some economy units that are only 1.5 hp. But all the serious ones are bigger. The high volume of water and heat is what blows the residue off the dishes. That means you have to cover light objects like silverware to keep them in the rack. Understand, that the scullery station has to be very though in precleaning. otherwise the wash sump will get loaded with food and the dishes will not clean properly. And the pump suction will get restricted form the screen getting loaded.
This covers both door machines and conveyor. Some of the high capacity conveyors have several tanks and wash pumps. Steam condensers, loading stations, and unloading stations. They also require a good crew to run them.
Now with your small undercounter, It's as simple as it gets. The trick is to have a good chemical supplier and soft water.