I don't mean Tibet as a political entity, nor do I mean Kashmir as a political entity. I know Ladakh is still fairly Tibetan in terms of cultural influences, but was the "Tibetan-ness" more spread out than that? How far back are we talking for Tibetan cultural movement into the area?
This question came about because a friend was telling me about the oral tradition of his particular group having come from Kashmir, and "India" before that, and while I'm highly suspicious of the accuracy of oral migration traditions in therms of broad place names, I actually don't know all that much about Tibetan culture in the Kashmir area. I'm much more familiar with more recent migrations into areas like the Brahmaputra Valley or settlement of groups like the Monpa in Tawang.
I've only encountered scattered bits of information about Kashmir in the literature I see out there so I'll give you my best educated guess to try and answer your question:
My own understanding of Kashmiri topics in general is quite limited, so hopefully an Indologist will come in at some point and correct me, but the area seems to have served as a highway for so long between Central Asia and the Subcontinent that it doesn't seem to me like there was a grander Kashmiri nation to speak of. Again, I could be wrong on that point.
As it relates to Tibetan topics, Kashmir shows up relatively early with reference to Padmasambhava^1 who is said to have been born in Oddiyana^2 estimated to have been in the Swat Valley of Pakistan, i.e. Kashmir. During the reign of Tri Song Detsuen, Padmasambhava became a great tantric master and was invited to Tibet to help establish the Dharma in the country and to sanctify Tibet's first monastery Samye^3. I'm telling this because it points to Tibet being Kashmirized, rather than vice-versa. This interesting paper actually discusses the roots of Tibetan tantra which can be traced through Kashmiri Saivism. Caution: while fascinating, it's not for the faint of heart. The Padmasambhava story is one of my favorites (I skipped the parts where he goes to Bhutan, seduces/is given the Emperor's wife for his own, freezes some archers, etc.) but in the terms of the question, it seems emblematic of the plantic of Kashmiri religious concepts into Tibetan soil. This was in the eighth century, Tri Song Detsuen died around 796 AD.
The Kashmiri-zation of Tibet, and then the Tibetanization of parts of Kashmir (Ladakh in particular, as you mentioned) probably continued in this way over the centuries, with Kashmiri Saivites feeding Tantric concepts and initiations to Tibetans who wandered into the area for business, trade, or religious reasons (possibly on pilgrimage to Oddiyana?). If you click through The Treasury of Lives, you'll stumble upon religious leaders, especially in the 1200s that were getting initiated into Tantric rituals and philosophy in Kashmir.
That said, outside of Ladakh, I don't know how Tibetanized Kashmir ever became, but my gut tells me "not very." For one, it's a pretty safe bet when the local language doesn't have their own term for it, that it's a foreign concept to them. I.e. In Tibetan, "Tibet" is "bod," "Bhutan" is "'brug yul," "Sikkim" is "'bras ljongs," India is "rgya gar," China is "rgya nag," and "Ladakh" is "la dwags."
"Kashmir" in Tibetan is "ka sha mir." I might be missing something, and it's possible that the region might have been subsumed under earlier polities (i.e. Zhangzhung) but it's notable to me how Tibetans today don't have a name for the area.
Shakabpa, famous as the first post-exile Tibetan historian, writes in his bod kyi srid don rgyal rabs (literally History about the Politics of Tibet):
de bzhin sbal ti si than yul skad l'ang bod skad dad ha cang 'brel che yod pa de yang lo brgya phrag drug pa dang bdun pa'i nang de dag bod kyi chab khongs su gtogs pa dang bod pa'i dmag mis lo mang sa srung du bsdad pa deb
Which is something along the lines of:
Similarly [to the connection between Tibetan and Burmese languages] there is believed to be a strong connection between the dialects of Tibet and Baltistan^4. As for that in the 600s and 700s those [the Baltisthanis] were subsumed under the dominion of Tibet and soldiers were stationed^5 for many years in that land to guard the border...
So what we have is reference to the area coming under Tibetan political control as early as the 600s (which would be under Songtsen Gampo, so again, we bump into the possibility that the land was previously under Zhangzhung until the Yarlung's conquered it), but I still find it a bit strange that Shakabpa has to use the term "sbal ti si than" to describe the region, which is clearly not Tibetan in origin, and he's in the middle of a description about where Tibetan language and literature comes from.
The History of Ladakh is another animal in itself. I'm not as well versed in its specifics as I am of Lhasa or Bhutan, though it's next on my list to conquer. I know A.H. Francke's A History of Ladakh and A History of Western Tibet are considered classics in this area. A lot of the Tibetology in that area of the Himalayas was done through the missionaries as they seemed the most academically focused until very recently. (Hell, our best Tibetan Dictionary still comes from H.A. Jaeschke who was one of them!)
Anyway, I'm sorry I couldn't be of more help. I'm in the middle of reading a bunch of other things right now, but Francke's work are on my summer reading list so I'll see if there's anything else I can find to help.
I never miss a chance to discuss naming conventions. "Padmasambhava" is Sanskrit for "Lotus born." In Tibetan that would be "pad+ma 'byung gnas" (Tib: པདྨ་འབྱུང་གནས།, Pema Jungney), literally "Arisen from a Lotus." He has Eight Manifestations, the most famous of which being Guru Rinpoche, which he is more commonly known among both the Bhutanese nationality and the Nyingma sect. This manifestation has come to be almost representative of his worship among the Nyingma has come to be a point of contention among Gelukpa-Nyingmapa tensions in the eighteenth century.
Oddiyana, also known as "o rgyan" (Tib: ཨོ་རྒྱན་, Orgyan, Ugyen) is a common name among Tibetans, and a name directly associated with Padmasambhava. Bhutan's first King was Ugyen Wangchuck.
Tibet's first Buddhist temple was built by Tri Song Detsuen's ancestor, the First Dharma King, Emperor Songtsen Gampo. He married two Buddhist Queens, one from Nepal and one from China. The Chinese Preincess, whom the Tibetans refer to as Kung Chu, used Yijing geomancy to establish the location for a temple, which his Nepali wife (so the story goes) Princess Belsa, designed and built according to Nepali customs. The Temple, the Jokhang, is considered Tibet's holiest site. Twelve more temples were built, the vast majority in Tibet, while two ended up in modern Bhutan. The difference is that temples are public places of worship, while monasteries are for monks to formally study. Their purposes and functions often overlap.
lit. yul skad = dialect, skad = language, so it's a bit weird here that he specifies "Baltistan dialect and Tibetan language."
The word here is "bsdad" which literally means "dwell," "stay," or even "sit." But I'm taking the responsibility for the word "station," which is, IMO, what you do when you place soldiers somewhere, as far as I'm concerned.