Is it considered a valid washing to place the forearms under running water and pass the other hand over them, or is it necessary to move the forearms themselves under the faucet?

The obligation is for water to flow over the entire limb, whether you move it under the tap or pour water over it with the other hand. The important thing is for the water to flow over the entire hands. And one must be cautious of obsessive whispering (waswasah).