Anything deemed "Mughal" is technically named after the Mongols, but only once that particular group came to India and set up shop there, so I don't think it would be fair to consider that a Mongolian origin.

Virtually all meals named "Mongolian X" are not Mongolian in origin, at least in the sense that they don't derive from the Mongolian heartland.

Let's take "Mongolian lamb" and "Mongolian beef" as examples. The core flavours and ingredients in these dishes involve sauces, spices, and vegetables that are not found in Mongolia itself and are not part of traditional Mongolian cuisine. Typically, they might include soy sauce, mixed vegetables, Chinese 5-spice, etc.. None of these are found in Mongolia or used in Mongolian cooking.

The names may derive from cooking traditions that developed out of remnant groups from the former Mongolian empire territories, but this is pure speculation on my part. Furthermore, many of these kinds of dishes are decisively Chinese in origin, and China is engaged in an ongoing and politically sensitive 'debate' over who owns which cultural traditions, complicated by the large number of ethnic Mongolians in China's province of inner Mongolia.

Mongolian Barbeque appears to have originated in Taiwan in the 1970s. So it has even less to do with Mongolian cooking than other dishes.

Traditional Mongolian cooking of meat is basically limited to boiling it, and occasionally stewing it, with either water or milk as the main liquid. They also developed a style of dumpling learnt through interaction with the Chinese.