We call it dishwashing, and the fluid is just called dish soap! :)

In the US, we ‘wash the dishes’ or ‘do the dishes’. There are different ways to clean them. I most commonly put the dishes in the dishwasher or load the dishwasher and then add soap. The added soap is called dishwasher/dishwashing soap/detergent. People without dishwashers wash/do the dishes by hand. This uses a different type of soap than a dishwasher uses. It’s most commonly called dishwashing liquid or dish soap. Note, there is also the term ‘hand wash’. In this context it refers both to dishes and to clothes and refers to something that won’t be washed by machine. Don’t put the steak knives in the dishwasher. They have wooden handles and need to be hand-washed. That pot doesn’t fit in the dishwasher. Wash it by hand. This blouse is delicate and must be hand-washed. (Unlike most of the clothes which are machine-washed)

We use 'washing up liquid' to 'wash the dishes' or 'do the dishes' :)

Cleaning agent,dish detergent,dish soap.

You wash the dishes with washing up liquid and a sponge 🧽 You can also say you need to do “washing up” after dinner. This is in GB English