The Dakini connection
Here is an extract of Chinese Magical Medicine by Michel Strickmann implying the direct connection between the Inari and the Dakini (p. 272).
The “Dakini” or “Dagini” for which this complex of modern
Japanese possession-rites is named is, of course, the dakini of older
Tantric sources, identified since at least the eighth century with the
fox-phantom. As we have noted, the Chinese linking of the voracious
and ill-omened dakini-demonesses with the fox was based on their
Indian associations with the jackal. In medieval Japan, it became
customary to identify figures from the complex Indian pantheon of
Tantric Buddhism with native deities (rather like the interpret at
ioromana in the world of late antiquity). Thus it was natural for this
interpretatio nipponica to see in the supernatural fox-dakini the
original of Japan’s own Inari, a god of rice cultivation often
depicted in vulpine form.
Now why the Dakini. The Dakini is a Tantric Buddhist monster. It appears in the Mahavairocana Tantra and he (the Dakini) is overwhelmed by Mahavairocana Buddha and (s)he rides a jackal. The text is well known in Japan (as Mahavairocana sutra).
Note that the Dakini is present in numerous Indian stories with lots of variants... LOTS of (including sex, appeance, attributes, malvolence). It is also known in Tibetan version. Jackal being unknown in China and Japan, they translated by another small canine fox.
Note also that the Inari appears to be from an old local Japanese tradition, it later been related to fox (around the 12th century) before joining more or then with the Dakini. The Inari also has the particularity to not be well defined, especially if (s)he is male or female. The blur with mysterious Dakini is quite obvious (Hence why be wary of my claim about one version specifically of the Dakini, or to compare too much the Inari and Dakini, or even Dakini and Dakini-Ten).
So the local deity Inari is identified by the Buddhist Indian Dakini to some extend. Here is the Japanese Dakini-ten riding her legendary fox (I pick this one because it is a gorgeous one, notice the small rice sheaf in her hand) (Note also it exists a male version of Dakini-ten, carrying a sword/weapon instead of a sheaf of rice):
Now regarding the Kitsune/Huli jing(princess Daji). Here is from the Cult of the Fox by Xiaofei Kang (p234, note 13):
The fox motif in stories of Reynard and the fables of Aesop have
appeared frequently in European paintings and architecture since the
13th... tales are found in all countries of Europe, including
Russia and the Near and Middle East. The motif of the fox wife [ie
Daji/Kitsune, note by me] wa salso found in Chinese and Japanese
folklore and accross the Berling Strait among the Eskimos in North
Tales of Reynard is a Medieval French book telling the adventures of Reynard. In modern French, fox is renard, and the novel is Roman de Renard, where the main character, Renard was a goupil (ancient French word for fox). The novel was so popular that foxes are now called renard and no more goupil.
Here is now a kitsune from a japanese temple:
Notice the kitsune DOES not have nine tails, only one tail on this statue. Notice also the sheaf of rice in his snout! In japan you will find kitsune garding Inari's temple. So don't assume the Chinese Huli jing and the japanese kitsune are strictly speaking the same creature. The kitsune is more than probably a port of the former, but it evolves quite differently.
Finally, there is no connection implied between India and Daji/Kitsune. They are from local Chinese tradition.
Here is now an example of a "true" Kitsune story (I mean by true, from japanese culture, not that you will find a kitsune as your girl next door). I push you first there, where you will find a resume of the original text as well as very beautiful original drawings.
See here some japanese warriors hunting an innocent two-tailed fox (The site I linked give you the Otogizoshi, browse it for the text and marvelous illustrations):
The story goes like that (short summary): Tamama-no-mae is the favourite concubine of Emperor Toba-no-in, reputed for her extreme beauty, marvelous scent as well as incredible cleverness. But unfortunately she is in fact a terrible kitsune.
Later tradition, not the Otogizochi, will link her to Chinese princess Daji, possessed by a fox spirit, due to the similitude in story. Now Daji story predating Tamamo story there is still a chance this is a port.
And here is the thing, all of that is buddhist in nature. And buddhism comes from India, so most of the time, you find those creatures (princess Daji, or Tamamo) mentioned as "coming form India" (Note that the Otogizoshi does not, she is Japanese, period). But one as to not see that as "coming from true India" but much more a way for Chinese and Japanese to mention something exotic.
Tamamo is still an incredibly popular character, you find her haunting dozen of modern stories, novels, mangas and video games:
Here is a VERY nice modern representation. I push you to notice the 9 tails. Modern japanese does not know their own folklore ;D