The cook takes it and sends the tDNA on its merry way, and then calls for the next ingredient. When the next ingredient arrives, the cook joins the new amino acid to the tryptophan in a growing chain of amino acids that will eventually be the root growth hormone. You're absolutely on the right track. Genes are completely transportable. If it codes for a protein we call insulin in people, it will code for insulin in plants, in pigs, in rabbits, etc.
Promoters can vary quite a bit in how general or how specific they are.
If you think about their function, you can see why this is true. The major reason that promoters aren't transferable is that not every plant needs to react to the same thing, and plants and animals certainly don't respond to the same thing. For instance, we must have a promoter that responds to food being chewed in our mouths, because when we start to eat, we produce more saliva so that the enzymes in saliva can start breaking down the starches in our food. We need that. But a plant doesn't eat.