In the old Bond movies there is at least some continuity with Connery/Lazenby/Moore, with something of a soft reboot after Moore leaves.
On Her Majesty's Secret Service is the one with Lazenby, and that came out between two Connery movies. It references Dr. No and From Russia With Love explicitly, and at the beginning of the next one Bond is out for revenge, presumably because of the death of his wife in the previous movie. At least two Moore movies reference his wife being killed.
Also, from Connery-Moore Bond's age is consistent enough. Moore was actually a little older.
I don't remember if the Dalton/Brosnan movies explicitly reference earlier movies with different actors, but I suspect that they don't.
Lois Maxwell plays Moneypenny through the end of the Moore era. When they get a younger Bond with Dalton they also get a new Moneypenny.
Dalton, Brosnan, and Craig each had their own Moneypenny, who was consistent through their run as Bond.
Q does stay consistent from the second movie through the Brosnan movies, with John Cleese being a different character when he replaces him (and they appear together in one movie).
There's Moore/Dalton overlap with one of the actors who plays M, so there's some continuity over that "soft reboot." I know Judi Dench is in the Brosnan and Craig movies, but it's clear when she showed up in the '90s that she is a new person in the role of M, and not the one Bond served under before. So it doesn't make sense to say that the character is the same person in the Brosnan and Craig movies if Casino Royal is Bond's first adventure.