Shinto has a complicated relationship with funerals.
First, death is a source of kegarë, pollution, and so people purifying themselves to take part in a matsuri were supposed to avoid the dead, and even the bereaved. Obviously, this creates a conflict if the priest is asked to perform a funeral.
Second, historically Buddhism appears to have taken over funerals very soon after arriving in Japan. In the Edo period (1600 to 1868, roughly), almost everyone was required to have a Buddhist funeral, provided by the particular Buddhist temple to which they were attached. This was part of the state project to ensure that no-one was a Christian. Even Shinto priests had Buddhist funerals.
The requirement to have a Buddhist funeral meant that those Shinto priests who rejected Buddhism, an increasingly large group as the Edo period progressed, wanted to get an exception. Obviously, they wanted to have Shinto funerals, but the two factors above meant that there wasn’t really an active tradition of Shinto funerals by that point, so they had to create them, and then campaign for permission to use them. By the end of the Edo period, most priests were allowed to have Shinto funerals, although other members of their families were not. In the early Meiji period Shinto priests campaigned for permission to perform funerals for everyone. After a few years, they got it. A few years later, the Meiji government decided to double down on “Shinto is not a religion”, and priests at important jinja were forbidden from performing funerals. After the war, with general religious freedom, Shinto funerals became possible again.
Thus, despite being much less common than Buddhist funerals in Japan, and having a somewhat strained relationship with Shinto theology, Shinto funerals have a great deal of symbolic importance for the Shinto community. They are given much more emphasis in literature and theoretical consideration than Shinto weddings, for example. Indeed, much more seems to be written about Shinto funerals, which serve a tiny fraction of the population, than about Shichi-Go-San, which a majority of children experience.
These days, Shinto funerals are held at commercial funeral halls, never at a jinja. (I’m sure there are exceptions, as always, but exceptions to this one are really, really rare.) The basic form of the ceremony is a matsuri, with the spirit of the dead person taking the place of the kami. Purification is performed, offerings are presented, and then the presiding priest recites a norito. This is formally called a “saishi” (“matsuri words”), but the structure is broadly similar to a norito. However, it also includes a section in which the life of the deceased is recounted, a bit like a eulogy at a Christian funeral. This section is still broadly in norito form, but it has to use a lot more modern language, even if the deceased was a Shinto priest. I imagine that a Shinto funeral for a computer engineer would have some very unusual terms in the saishi.
After that, all the attendees pay their respects in almost the normal way. Offering a tamagushi appears to be common, and then you bow twice, clap twice, and bow once. An important difference from a normal jinja visit is that, at a funeral, you clap silently. You bring your hands together, but do it slowly, so that it makes no noise.
As for what Shinto has to say about what happens next to the deceased — that is a topic for another post.