This new ideas seem to be spinning fast and furious here. My two cents.
First absorption of the agricultural practices would almost certainly be through the Tlinglit. They were not only the furthest north, but they were the regional trading power, and had access to at least the Yupik IOTL. Although it's difficult to be sure with oral histories, they seem to have lived in the area for centuries, and were almost certainly already there in 1400.
There's a couple of unusual things about Tlingit culture. They were split into two moieties - the Raven and the "not Raven" (Wolf or Eagle, in different places). One was only supposed to marry from the opposite moiety. Within each of these groups, there were "houses" with their own noble lines. However, noble authority is only valid for those of that bloodline - authority is genetic, not territorial. As a result, settlements had many parallel lines of authority. One thing that is important for contact with the Thule is that ancestry, including nobility, is figured through the mother, not the father. Typically when marriages were arranged, men would move into the community, and there was nothing untoward about a non-Tlingit marrying in.
Another thing to consider is the Tlingit notion of property was well-developed, to the point where they not only considered property to be land or objects, but ideas. Thus one could have ownership over a dance, story, or song, somewhat akin to the modern Western ideas of intellectual property. For this reason, it's highly likely that the first Tlingit to take up agriculture will see the practices as clan property, and be very hesitant to let other clans in on it unless some advantageous trade can be made.
I'll leave the steps of the process of agriculture to Pesterfield to discuss, since he's doing a pretty good job. I'll briefly outline the stages I think Tlingit culture would go through, however.
First, there would be the initial introduction to agricultural practices. This can come either piecemeal through trade and trial-and-error, or in a more comprehensive fashion as bachelor Thule migrate into the region and marry in. Although the first clans to establish agriculture would attempt to keep this under wraps, the marriage system, where young men leave the clan of their birth and often move to a distant place, would mean that knowledge of the practices of agricultural husbandry would spread far and wide.
That doesn't mean that application would be universal, however. Except for low-level petty warfare, Tlingit life was pretty nice, all things considered, and few people went hungry. Adoption of agriculture initially may be seen as useful not because of the extra calories it brings in, but because it seems a long-term time saver compared to the historical practices of gathering wild plants. Also, the traditional diet was low in some nutrients, like Calcium and Vitamins A, C, and D. In order to compensate, Tlingit had to eat a lot of offal (as the Inuit did), but with access to more plants to provide trace elements they can have more of a protein monoculture, and should be more numerous and healthy.
Before the Thule get heavily involved in the region, I expect the Tlingit would begin expanding outward due to population pressures. Although with access to the Thule agricultural package, they could in theory climb up the slopes and attempt to displace the inland groups they traded with, given their maritime I think they'd turn against their southern neighbors. Presumably, they'd move into the territory of the Nisga’a and Tsimshian. The latter in particular is important, as it provides access to the Skeena River. This is the first area that would be considered "arable" in part under western terms, and more amenable to forming an agriculturally focused nation than the thin array of coastline the Tlingit have been limited to so far.
From here, there's breaks in the mountains which would lead to the Fraser Valley - the single largest agricultural area in modern BC, which is known for having huge amounts of wild...arrowhead! While I'd expect further coastal penetration as well, the main thrust may be down the Fraser and into OTL's Vancouver area instead, as the Dene groups inland will probably be pushovers in comparison.
As others have said, in the long run the Thule will probably decapitate the Tlingit nobility. Hell, the system lends itself to it - Thule men could go in, kill off most of the males in the clan, marry the widowed women - and their children will have legitimate clan status!
I'd expect that overall Thule genetic influence would be more minor than one would expect. Given the Northwest Coast is going to be more agriculturally productive than the Thule realm ultimately, it won't be long before the population is denser than the core Thule territory. Thus, which ever lucky Thule groups establish rule will probably find it pretty easy to resist further invasions after a few generations - meaning there will not be wave after wave of Thule migrants entering the region. Probably some Tlingit-descended groups will keep ahead of the invasion wave, particularly inland, and continue to expand until Eurasian plagues intervene.
As to the cultural drift thereafter, it's difficult to say. I'd argue at least for the sake of the story, however, that the analogue should be more Norman England than Anglo-Saxon. In the first few generations, there would be a recognizably Thule ruling class, but as time passed, even Thule-descended nobility would turn to local languages, except when they had to parlay with "Thule" elsewhere.