While similar theory has applied to other contexts, the notion of "gunpowder empire" is something that is usually specifically used in a Central and West Asian context. In particular, the idea is that the three major gunpowder empires are a sort of successors to the Central Asian nomadic Empires that had previously dominated this part of the world, especially in the centuries following the formation of the Mongol Empire and its many successor states.

Essentially, in the pre-gunpowder world, the sedentary agricultural civilizations were vulnerable to attacks by Central Asian horsemen using bow and arrow. Thsi allowed them to retain military dominance over large stretches of central Asia. The Timurid Empire, which ended in the early 15th century, is often regarded as the last major nomadic empire, although significant Nomad states would remain on the steppes until the 1700'es.

The idea is then that the advent of gunpowder changed this dynamic. Gunpowder concentrated power in the hands of those able to equip and maintain large standing armies, and the ones who could handle the logistics demands of keeping them supplied. This did not favor Central Asian nomads, but instead strong, centralized state powers.

It is worth noting that the theory of gunpowder weapons leading to increased centralization is not limited to only Western and Central Asia. It is also a common theory in development of strong central states in Europe, seen best in the expansion of absolutism at the expense of a strong nobility. Same theory has also been applied to Japan, that saw the development of a strong centralized state after the arrival of gunpowder in the 16th century. It can also be applied in much smaller and more limited contexts - the 19th century Musket Wars in New Zealand for instance.

But the term gunpowder empires are generally not used in these contexts. It was developed for a West, Central and South Asian context, to describe the rise of the three empires you mentioned, and their replacement of the nomadic horseriding empires of previous centuries.

It is worth noting that some scholars have in recent years expanded the term to other contexts. Tonio Andrade has used the term to describe Qing China as well as Muscovite Russia in the 17th century, seeing as they came to dominate large stretches of Central Asia on the back of gunpowder technology.