Do we have any historical evidence that Jesus visited India? What is the evidence and the source for the belief held by many people in India that the Roza Bal shrine is the tomb of Jesus?
The recent written source for the idea that Jesus is buried at Roza Bal is here: Jesus in India - by Mirza Ghulam Ahmad - The Discovery. No forensic evidence seems to be cited. Ahmad relied on his own personal research of the subject and older Islamic traditions, oral and written.
As for Jesus visiting India during his life, See: Nicolas Notovitch: Notovitch is known for his 1887 book claiming that during his unknown years, Jesus left Galilee for India and studied with Buddhists and Hindus there before returning to Judea, although Notovitch himself (based on the wiki - I have not read the book) never claimed Jesus was buried in India or Kashmir but traveled there and then returned to Palestine. (Thanks to @LennartRegebro for this reference.)
But Notovitch is a problematic source: He later admitted that his story was essentially a fabrication, and as a European writing in Europe, it's unlikely that he was the source of those beliefs you refer to. Perhaps the writings of Swami Abhedananda: Although Notovitch had been discredited in Europe, Swami Abhedananda wrote a book called Journey to Kashmir and Tibet in which he stated that had visited Hemis and seen a document similar to the one Notovitch had mentioned, and produced a Bengali translation of parts of it, are better sources for this idea.
Regarding the validity of the claim that the Roza Bal Shrine is Jesus's tomb: In light of the extensive biblical testimony that Jesus was entombed in ancient Palestine, Resurrection of Jesus, Burial,Resurrection of Jesus, Tomb Discovery, one would need to explain how Jesus's body ended up in Kashmir. But if we accept the biblical account of the Resurrection, it is possible to say that once he was resurrected he could have ended up anywhere. But again, this is contradicted by the biblical account: His final appearance is reported as being forty days after the resurrection when he ascended into heaven[Lk. 24:44–49] where he remains with God the Father and God the Holy Spirit until the Second Coming of Christ.
Whether or not we accept the biblical account of the Resurrection, the numerous biblical references and their persistence would seem to put the burden of proof on Ahmad and his sources, and the evidence does not appear to weigh in their favor, particularly since we have other sources besides the New Testament attesting that Jesus lived and died in Palestine. If so, a very good explanation must be offered as to how his body ended up in Kashmir, thousands of miles away.
There is no historical or primary textual evidence that I am aware of which stated that Jesus ever traveled to India. I heard and read about this claim over the years and it seems to be more mythical and less historical. Now even though there is no major historical or primary textual evidence proving that Jesus traveled to India, does not entirely exclude the possibility that Jesus traveled to India.
If one looks at the 4 Gospels in terms of Jesus' travels, we are told that his travels were regional, that is to say, not beyond the Middle East. In contemporary terms, Jesus lived and traveled in present-day Israel, the Palestinian West Bank-(then called, Judea and Samaria), Jordan, as well as Egypt. It is possible that Jesus may have traveled to the town of Tyre in Southern Lebanon or perhaps around Southern Syria-(which was in fairly close proximity to the Galilee). However, the 4 Gospels, to my recollection, do not mention other or additional lands that Jesus traveled to-(besides the Middle East region).
I don't remember any the of Apocraphyl stories mentioning Jesus' travels beyond the Middle East as well; nor is there any mentioning of Jesus traveling to India-(or other lands beyond the Middle East) in The Letters and Acts of The New Testament. Flavis Josephus' short discussion on Jesus, to my knowledge, does not contain Jesus' travel itinerary. These are some of the earliest historical sources we have on Jesus and all of them were written a generation and generations after his death.
Admittedly, I am not well versed in the Ancient Literature of India, which may have some interesting primary documentation proving such a claim, though I am currently unaware of such texts and sources.
As to whether or not Jesus is entombed in the Roza Bal Shrine, it is similar to claims that Jesus is entombed at the Talpiot Cave in Jerusalem or that Jesus was originally enshrined in the Garden Tomb in Jerusalem as well. And of course there is the Church of the Holy Sepulcher in Jerusalem which has had its own lore for nearly 2000 years. In other words, the enshrinement and entombment of Jesus has produced-(and I expect will continue to produce), more stories and extraordinary claims thereby furthering its lore and mystique..........which is outside the realm of history and archaeology.