It's excruciating to read stuff about Judge Kavanaugh as an "originalist," a "textualist," a "conservative in some political theory sense." You want to know how he's going to vote as a Justice over the next few years? (The time qualification is relevant.)
The first question to ask is whether there is a mainstream Republican conservative view on the issue presented. (a) If there isn't, we have no way of knowing what a Justice Kavanaugh would do. (But, if there isn't, not many ordinary people are going to care. I have a professional interest in seeing how Justices deal with the question of whether a fish is a tangible object, but no sensible person without such an interest cares.) (b) If there is, Justice Kavanaugh will find the mainstream conservative view compelled by "the law." The only interesting question will be how emphatically he talks about compulsion.
The next question, though, is whether there's a difference between the establishment Republican conservative view on the issue and the Trump-Republican view on the issue. If there is, a Justice Kavanaugh will probably find the establishment Republican view more persuasive than the Trump Republican view, though his opinion, if he writes one, will make some concessions to the Trump Republican view (on the implicit theory that Trump Republicans might become the establishment over the next few years). The distinction the Chief Justice drew in the travel-ban case between "this President" and "the Presidency" will have some weight in Justice Kavanaugh's thinking.
Over the course of his time on the Court, the issues of interest to conservative Republicans will change. (There will be a conservative Republican view about legal regulation of artificial intelligence, but we have no idea what that view will be.) The chances are high that even as the issues change, Justice Kavanaugh's positions will not change relative to that of then-existing conservative Republicanism. (Ideological drift, the weaponization of the First Amendment, and all that.) But there's some chance that on some issues he'll be stuck in establishment conservatism circa 2010-15, and might occasionally become a "maverick."
Not for a while, though. Here's a thought experiment/prediction. Suppose Democrats manage to win enough elections to enact Medicare for all in 2022 or 2026, over concerted Republican opposition. Justice Kavanaugh will hold that, though there are constitutionally permissible ways to enact Medicare for all, the one the Democrats chose was unconstitutional on some yet-to-be-devised (but probably lurking somewhere in the Heritage Foundation's papers already) constitutional theory.