I understand that answers will be even more speculative than is normal for ancient history, due to the paucity of sources. When I ask a question that seems to be looking for concrete answers, I just mean to ask for prominent theories in the field.
From what I understand, the canonical story of Celtic is that what we could call the original Celtic cultural-linguistic phenomenon originated in the Hallstatt cultural area in the early-mid 1st millenium BCE, then spread out from essentially what is now Bavaria to cover huge swathes of Europe. What are the most prominent theories for the cause of the expansion of Celtic language/culture? To ask about a particular case of expansion, what would cause a group of Celts to upend themselves, go all the way to Greece to raid/settle, and then eventually move over into Asia Minor to finally settle down? Would the founding population of Galatians have mostly been a warband of men that intermarried with locals, or would there have been mass migration of women and children comparable to the number of men?
Centuries later, the original Celtic area would become Germanic - how did this process occur? What are the prominent theories regarding the pre-Migration Period expansion of Germanic peoples/cultures in general? The expansion from the homeland along the North Sea and the Baltic across Central and Eastern Europe is comparable in spread to the earlier Celtic expansions. What do we know about the replacement/assimilation of the Celtic populations in Pannonia and the Balkans, re: when and by what other peoples?
First of all, before we can think about answering the question, there's an awful lot to untangle with the word Celtic. It can mean one of three things, two of which (at least) you've conflated in your OP. Not a problem, but unfortunately a very common misconception. Celtic can refer to either a linguistic group (someone or people who spoke a language belonging to the Celtic languages), a particular style of material culture (i.e. objects which are similar in design and influenced by one another) or as part of an ethnic group (i.e. a group of people who identified as 'Celts').
Celts, as a self-identifying ethnic group, did not exist in antiquity. Keltoi (and variants) was the word used by Greek (and later Roman) writers to describe the various groups of 'barbarians' who lived to the north of the Mediterranean, in what would now be called Central Europe. (Edit: Although the term was not, by any means, used consistently by ancient writers - see /u/Tiako 's post below.) These people wouldn't have recognized any common identity or felt part of any larger cultural or political unit.
Celtic, in the material culture sense, seems to be what you're (primarily) asking about. The material culture which archaeologists have dubbed 'Celtic' first appeared in the areas just to the north of the Alps (modern Switzerland, Austria and Southern Germany) around 700 BCE and spread and evolved throughout Central Europe, Western Europe and the British Isles, with thousands of local variations, evolutions and styles. There is no evidence that these styles of metalwork were spread via invasion or migration - rather, they spread as fashion still does today: through exchange, local knock-offs, fusion of styles and appropriation.
The Celtic linguistic family was widespread in antiquity, and consisted of a range of languages which would not be mutually intelligible but which historical linguists can see possessed similar roots and common grammatical features. Modern Irish, Welsh and Scots Gaelic are the descendants of the Insular branch of these languages. In the past, archaeologists assumed that everyone who used Celtic style material culture must have spoken a Celtic language, and that since these objects originated in the Northern Alps and spread (roughly) westward, that the languages must have spread that way too, through invasion and migration. Now, we know that this is bunk, because why should having a fancy La Téne-style sword mean you have to speak a Celtic language? So there is no intrinsic connection between 'Celtic' material culture and Celtic languages, although we know that there were certainly times in history when Celtic-speakers used or manufactured some styles of 'Celtic' objects. We know that Celtic languages were spoken in the British Isles, Gaul, Iberia and in Northern Italy, and (if we can believe Pausanias' reports about certain Galatian words) it was probably spoken further east too. The origins of the Celtic languages, however, are disputed. The most recent theory, put forth by Barry Cunliffe, is that proto-Celtic emerged as a lingua franca along the Atlantic seaboard of Europe, and spread westward eastward, but this has yet to gain widespread acceptance.
So, in short, the 'canonical story' of the Celts is really nothing of the sort. Celtic languages and 'Celtic' material culture are two separate, if overlapping, phenomena, neither of which has any evidence to suggest that it was spread through migration or invasion.
That is not to say, however, that migrations and invasions by Celtic-speaking peoples did not take place in antiquity. You mentioned the Galatians specifically: according to Livy (Book 38, ch. 16) they were recruited by King Nicomedes of Bithynia from three Celtic tribes living north of Greece to fight a war in 278 BCE. The three tribes in question, the Tectosages, Trocmii and Tolistobogii, decided to bring their families across too, and eventually settled in central Anatolia, raiding their neighbours and generally causing trouble for the rest of the century. They ruled over the native population, however, rather than displacing them entirely, and although they kept some distinctive cultural features, seem to have become Hellenized by the 1st century BCE at the latest. There seems to have been a major movement of Celtic-speakers into northern Italy at the end of the 5th century BCE, which brought them into contact with the Romans and ended with the famous sack of Rome in 390 BCE. As late as the 59 BCE, when the migration of the tribe of the Helvetii was used by Caesar as the pretext to begin his Gallic campaigns, migrations by Celtic-speaking peoples were happening for all sorts of reasons. But none of these population movements were the reason for the spread of either Celtic languages or 'Celtic' material culture.
The Germanic migrations at the beginning of the Middle Ages are much simpler (though it's a relative term!). These were migrations of primarily Germanic speaking tribes into a weakened Roman Empire. There seem to have been two major reasons. Firstly, pure opportunism to take advantage of the wealth of a crumbling Roman state. Secondly, there seems to have been a knock-on effect caused by the migration of nomadic groups from central Asia, above all the Huns, over the course of several generations, which sent other tribal groups into the Empire as refugees. I only know the basic outline of the migration period - hopefully someone else with a better knowledge of Late Antiquity will come along and answer in more detail!