We have many recent examples of movement of peoples. For example, Indian settlement in Fiji, European settlement on Australia and New Zealand, and before that multiple waves of European settlement in the Americas and South Africa. We even have examples in progress today, such as Han Chinese settlement in western China and Javanese settlement in non-Javanese Indonesia.

These movements are not mysterious because we know what happened - these events are recent and well-documented. It's harder to track early movements, especially prehistoric movements. We can try to track movements of and changes in technologies and cultural practices such as burial practices via archaeology. However, technologies and cultural practices can be adopted by neighbours and spread without movement of peoples, and peoples can move and adopt the technologies and cultural customs of their new homeland, so archaeological information is often ambiguous.

We can also try to track the movement of peoples via ancient and modern languages. Where ancient languages were written, and some writings have survived, even if very few, it can be enough to identify language families, providing clues to movements. Where there are no writings, place names can preserve elements of language, and also personal and tribal names recorded by literate neighbours. Otherwise, the modern distribution of languages and their relationships provides clues to past movements. Again, languages can move without peoples moving, and peoples can move and adopt new languages, so there are limits to this. Languages do seem to diffuse less and be abandoned less often than technologies and burial customs, so provide excellent support for archaeology.

Finally, genetics provides excellent evidence for the movement of peoples. Much better than technologies, cultural elements, and languages, genetics provides a good measurement of the mixing of peoples. The genetics of modern populations can be readily studies, and provides clues to past movements, and the study of ancient DNA (e.g., from burials) can give excellent information about the timing of movements of peoples.

Before the technology for modern genetic studies was available, what was there? In some cases, historical records. These sometimes took the form of "the Huns came from the east" and give little concrete information other than timing. In many cases, especially older cases, there are no written records. Sometimes, oral traditions were written down later, and provide some clues, but are rarely precise enough to allow movements to be tracked. The ambiguity of archaeology has already been mentioned. This meant that often the best clues were languages. Thus, it appears that there have been expansions of Slavic peoples, Indo-European peoples, Turkic peoples, and Bantu peoples, to name some prominent examples. The modern distribution of languages provides reasonable clues as to where such peoples spread from, and sometimes an archaeological culture can be associated with their homeland. In many cases, there is significant uncertainty about the details, but a consensus as far as the big picture goes (e.g., Slavs, Indo-Europeans, Anglo-Saxons, Turkic peoples).

In other cases, there are still mysteries, e.g., the Huns. The Huns demonstrate some of the difficulties very well. The Hunnic language is almost completely unknown. There is no known distinct Hunnic archaeological culture - Hunnic material culture on the steppes was probably very similar to that of other steppe peoples, and we know of no reliable markers to distinguish Huns from other steppe peoples, and Huns in Europe probably adopted many elements of the material culture of their neighbours. Written records by their neighbours don't help us learn the origin of the Huns. Linguistically invisible, archaeologically barely visible, and on the edge of written sources, we lack the things that let us track the movements of peoples such as Slavs, Turks, etc. There are other cryptic movements of peoples, e.g., the Basques, in whose case we know their modern language, but the lack of known related languages doesn't allow us to make use of its relationship with other languages. In this case (and in others), possible evidence of early movements of peoples has been obliterated, or at least greatly obscured, by later movements of peoples. A final problem: sometimes what written sources are available are misleading. Written sources can present origin myths as reliable fact, e.g., the Goths, as discussed by u/shlin28 in https://www.reddit.com/r/AskHistorians/comments/5wpwe1/ancient_goths_what_was_their_whole_deal/