While I've heard that the Sami domesticated Reindeer and that they did so as early as 3000 years ago, I can't seem to find any good sources on the how. Its a large, powerful animal capable of huge leaping bounds that can run far faster than you -- How would you go about getting such a critter to a state where you can ride them, milk them, etc?
Weirdly enough I actually took a course that focused on herding and we spent a good chunk of time learning about the Sami. So my apologies if my flair does not match the topic!
Herding in general is a sort of abstract approach to domestication. What you have is a large group of animals who have a hierarchy that allows for a leader to essentially direct the group. Herd mentality, literally. Herders rely on this to assist in moving the herd when needed, you can direct the group should you have the manpower to direct them along the landscape. Through this you get this weird middle ground between domestication and hunting/gathering. You are constantly following the heard and attempting to direct it in a direction, but at the same time you are also selectively breeding the herd for a desired set of traits. Part of this is due to the sheer footprint of a herd, they need miles and miles of room for grazing to sustain a large population. This means that you will be moving around a ton, and most likely not keeping the herd in one location like one would do for an animal like a pig or goat.
Domestication then comes with time. Groups may splinter off, some animals may have a better temperament, or even just pure luck. At some point people would become able to work with the animals a little more directly, allowing them to be used as beasts of burden or for milk production. This would have further sped up the domestication process as you would be selecting the animals with a temperament that allowed for close interactions, similar to dogs from wolves.
As for the timing, remember that humans had been hunting herds of large animals for thousands of years. Over time they would notice the ability to direct the herds, and the rest would be left to time as they became more skilled at managing the herd.
In the end of the 10th century, a Norwegian chieftain, called Ohthere (usually rendered Óttar in Old Norse) took a visit in the court of King Alfred of Wessex (d. 899), and told a bit about his homeland, including his relationship with the hunting gatherers (the Finns.....not necessarily, or exactly corresponding with the inhabitant of Finland) in the far north.
In this alleged dictated description, Ohthere narrates about his 'wealth' in his farmstead in northern Norway:
'He was a very prosperous man in respect of those possessions that their wealth consists of, that is, of wild animals. When he sought the king, he still had six hundred domesticated animals unsold. These animals they called reindeer (hranas); six of them were stæl reindeer. They are very valuable [prized?] among the Finns (Finnas), since they [the Finns] catch the wild reindeer with them [stæl reindeer]' (Bately ed. 2007: 45f.).
This is the first written account on the reindeer (this modern word itself only appeared in lexicography in the 15th century) in (historical) English!
The interpretation of OE word stæl is certainly problematic even among the researchers, but it is usually annotated as a kind of 'decoy/ bait' animal to lure a group of wild reindeer in front of the human (Storli in Bately ed. 2007: 94). Alternative interpretation is a milker or a leading animal, but they could also sometimes be functioned as a kind of decoy as well, such as, in the breeding season.
I also wish to add a brief note that the scale of reindeer herding among the Sámi people seemed to have been relatively limited until about 500 CE (Early Iron Age), so, 'the 3000 years of history in their reindeer herding', as mentioned in OP, might be a bit exaggeration (Hansen & Olsen 2014: 86-88).
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