You have two questions in here: one about meat, and one about meat delivery vehicles. I'll answer each in turn.

First, the meat. The vertical, rotating kebab was invented in Bursa, an Ottoman town near Istanbul in the early to mid 1800s. If you go to Turkey today, you'll still sometimes see them doing it the OG way, cooking the meat with charcoals inserted into a little open shelf system next to the kebab instead of with the electric heaters they use today. That meat in Turkish is called döner kebab, which means something like "always turning kebab" because of the verb tense used. That phrase got borrowed into the Arab provinces of the Ottoman Empire with another Turkish word for turning, "çevirme/ chevirme." But Arabic doesn't use the "ch" sound, so it got adapted into that language as "shwarma." The Greeks probably used a Turkish word for awhile (Bursa had a very large Greek population until the early 20th century), but eventually switched to a Greek translation for "turning/rotating," "gyro," which you'll recognize as the same root for things like gyroscope. I suspect, as is often the case, nationalism had a role to play in this, like how Greeks renamed Turkish coffee to Greek coffee after the 1974 Cyprus Crisis.

Here's where the question gets tricky. That meat style actually got imported to Mexico City by Syrian and Lebanese immigrants, where it is called "al pastor," or "shepherd style." Why they chose that term, I don't know! In Mexico City and a lot of northern Mexico it is still done up on the rotating kebab rack, although in the US most of the time it is just marinated and grilled pork because they don't have the specialist cooking rack. It is great either way!

The Mexico connection for the meat makes the next part harder to answer. In the regions of the former Ottoman Empire, everyone uses a version of flat bread with a pocket in the middle we know in American English as "pita." Everyone there also uses a wide, wheat tortilla-like thing called lavash. Both bread items have varying names depending on where you go in Turkey, let alone if you go to Greece, an Arab country, the Balkans, or Armenia. A lot of that pita doesn't actually pop open and hold its shape well enough to hold stuff inside it, in my experience, because of different baking styles and probably different varieties of wheat being used. In Turkey, you will usually get a döner on a regular roll or wrapped in a small lavash. An important part of the story is the massive wave of Turkish immigration to Western Europe in the second half of the 20th century; döners you get in Western Europe are different than those in Turkey. Turkish immigrants started whipping them up in Berlin, Amsterdam, and London with local ingredients in the second half of the 20th century and give us something a little different than what you would get in Turkey today. This includes the meat stuffed in a pita, a huge helping of tzatziki, which exists in Turkey but doesn't usually find its way onto a döner, and a lot more vegetables. Its possible they were using a different, more gluten-rich type of wheat which allows for better pita-stuffing properties. Anyways, it seems to me, although I can't confirm it, this Western Europeanized version merging Turkish and European tastes is ancestral to what most of us call a "kebab" in America or the EU today instead of the döner you would get if you went to Istanbul today.

Bread has another part of the global, transnational story we are telling here, because Mexico is obviously the home of the tortilla. Corn is originally from Mexico/central America, and they have a million ways of preparing little corn cakes and wraps. I emphasize little, because corn has way less gluten than wheat, so you can only make your corn tortilla or sope so big. This is why a proper taco is so small. Mexican culture is fairly heavily split between the more Native American south and the more European immigrant influenced north. Mexico City and below will get you food with a heavy does of indigenous cultural practices and tastes with only a light European influence. Up north, where European immigrants (many from Central Europe) set up large ranches, the food is a lot like central and southern Mexican food with more European preferences imbued into it, kind of like an ancestral Tex-Mex. This is also why northern Mexican music is functionally polka and all Mexican lagers taste suspiciously like something you would get in Prague or Vienna. Anyways, northern Mexicans preferred making tortillas out of wheat instead of the proper corn. You can make a wheat tortilla waaaaaay bigger than a corn one because of all that gluten. The heavy European influence in northern Mexico and the southwestern American border lead to a whole style of cuisine with wheat tortillas instead of corn, and burritos came out of this, because a wheat tortilla can hold way more ingredients in it. Burritos only started showing up in central and southern Mexico, as far as I'm aware, after American tourists started showing up and asking for them there.

Now, as for similarities between lavash and wheat tortillas? I'm a native Californian who got into Ottoman history and moved to Istanbul at one point. It blew my mind that you can get the exact same quality and style of store bought wheat tortilla in the exact same plastic packaging with the zip-lock top in Istanbul as you can in California. Only in Istanbul, they are labelled "lavash." I know Askhistorians is not really the place for speculation but... I highly suspect that in the post WW2 industrialization of Turkey along with ever increasing ties to American industry, someone realized how similar lavash were to tortillas, bought an industrial tortilla making and packaging machine (or several!), imported them to Turkey, and started churning out "lavash" packages. You can still get the (much) larger traditionally prepared lavash, but most supermarkets carry the suspiciously North Americanized variety. That story seems to be a fun one of parallel evolution in meat delivery vehicles that ultimately crossed together in the late 20th/ early 21st century because of globalization.

So: the meat has a shared history. The meat delivery vehicles are more complicated, although in both the cases of the Europeanized döner and Europeanized tortilla have some similarities. We also see a convergence of meat delivery breads because of globalization. I hope this helps! Hopefully someone better versed in Mexican history can help clean up my sketching of Mexican food development.

Edit: I led with a sentence fragment on accident, which bothered the hell out of me. I fixed it!

I just want to add to /u/PaxOttomanica’s answer, to extend the discussion slightly. While gyros, döner, and shawarma all ultimately date to the modernization of spit-roasted çağ kebabı, the burrito points to a larger pattern because, before the convergence of industrial processes (/u/paxottomanica’s theory is possible), they definitely existed as separate traditions.

To make a long story short, our DNA makes things with amino acids. Some amino acids our bodies can produce, but eight or nine amino acids our bodies can’t produce. These are called “essential amino acids”, and we have to get them through our food. A protein source that contains all of these essential amino acids in high amounts is called a complete protein (there are better ways of thinking about this, but this is the simplest one). Proteins from animals (meat, fish, eggs, dairy, etc) tend to be complete while proteins from plants tend to be incomplete, meaning they lack at least one of the essential amino acids in sufficient amounts (there are exceptions, like soy, quinoa, buckwheat, spirulina, but they tend to be rare globally). This makes sense: animals are closer to us, and so have similarly biological needs.

Different types of plants, however, tend to be missing different types of amino acids. Here’s the thing: eating meat was historically rare, especially for peasants who made the majority of everyone in non-hunter-gather cultures from the advent of agriculture until today. How did they survive? In general, you’ll see one of three things happen: the first option is to rely on either transhumant herding or fishing to get animal protein. The second option is to have a staple starch supplemented by some other animal protein, either dairy or eggs. The Irish diet, for instance, relied heavily on potatoes and butter/dairy or oatcake and butter, depending on the period (see this older thread or this one). In other areas of Europe, you might have diets based largely on bread (made from wheat, barley, rye, oats) and another source of dairy such as cheese. Tibetans likewise rely heavily on dairy, especially yak butter. The third option is to mix multiple plant sources. In areas without lactose tolerance, agriculture often featured prominently a cereal (wheat, rice, corn, barley, sorghum, millet, etc) and a legume (beans, peas, lentils, chick peas, peanuts, etc.). Think of a beans and tacos, rice and beans, hummus and pita, peanut butter and jelly sandwiches, etc. These are all examples of the third option.

In most places that had agriculture, typically a carbohydrate-rich cereal (wheat, rice, etc) or a tuber (potatoes, sweet potatoes, yams, cassava) provided most of the energy (calories) needed in a day. Alone, grains do not provided a sufficient protein balance (though they are not particularly high in protein, potatoes and some often do contain a good balance of essential amino acids; however, they contain insufficient amounts of two essential fatty acids meaning they, too, have to be supplemented, but let’s not needlessly complicate this conversation). There are a variety of ways to turn the dominant cereal into bread, noodles, porridge, or even just individual edible grains (like rice or bulgur), but simply having a grain alone won’t be enough because it lacks sufficient amounts of lysine, an essential amino acid. There needs to be another common thing else to supplement the cereal, either a legume (which have enough lysine but not enough methionine, which luckily cereals do have) or some animal produce or just a rich palette of other fruits and vegetables. This website, for instance, gives three sample Medieval European peasant diets: convenient for us, in one, the grain is supplemented primarily by cheese, in one primarily by beans, and in one primarily by meat, which I think gives a sense of the possibilities for diets in Europe and the Near East.

So, in one sense, the burrito and the döner-gyro-shawarma family raise in two different contexts, but, in another sense, they rise from the same basic need, just like rice and beans or peanut butter and jelly: to supplement a staple grain that historically provided the lion’s share of a peasant’s daily calories with an adequate source of lysine, an amino acids that grains lack in sufficient quantities and that the body can’t produce.