Down here they can be
* Shawerma
* Doner Kebab
In either case they are flat bread like pita or lebanese, with meat, onion (red with sumac), lettuce, tomato, and sauces (garlic yoghurt and hot chilli are my faves).
Gyros/Yeeros are typically made with what's more like a big, not sweet pancake (not as dry as a tortilla)and assembled as above.
Lamb, beef and Chicken are common...chicken not so much when I was a teenager but common now, or Fallafel.
Greek Souvlaki is similar to the pancake/tortilla but different spices, and more commonly lamb.
The meat on a spit is stacked slices and mince (ground) usually, except for the Souvlaki.
As a family, we usually buy a takeawy without wrap, and eat it like a salad, with a little of our own supplied flat bread...a full Kebab is too much for any of us.