A lot depends on how much of the essences of the character remains intact.
For example, Bond is - at base - a secret agent performing acts of mayhem for Britain (cheers, cheers) and defeating secret agents performing acts of mayhem for other powers (boo, hiss). He can be updated as the years roll on, with his enemies shifting from Russia to terrorist groups like Spectre and AQ. It doesn't matter, at base, if his stories are set in 1960 or 2021. What matters is the role Bond plays. The same can easily be said of the Doctor. By the time the First Doctor passes on, the character is firmly established as a time-travelling meddler and as long as that is there, the rest is just gravy. Of course, this does depend on both good actors and good scripts. Colin Baker and Jodi Whitaker both had problems with those and their doctors suffered, particularly when Chibnal decided to take a wrecking ball to the franchise.
A slightly wider franchise opens the range for newer actors. Star Trek could (and did) handle a captain who was not Shatner. The role of captain wasn't defined by a single person or persons, allowing Sisko and Janeway to have a shot at it. Archer was a weaker candidate and Burnham was not, at least at first, a captain in her own right. Rather the opposite. Nor was Green Lantern ruined by Hal Jordon being partnered by Guy Gardner and John Stewart, although things wobbled a bit when Hal turned evil and Kyle took his place. There's a place in the lanterns for Simon Baz and Jessica Cruz and no reason, at least on the surface, why they shouldn't have a chance. There's no rule that says Lantern #2 has to be a carbon copy of Hal Jordon (or Alan Scott).
Really, though, it depends on the character. Kyle suffered because he got a hell of a lot of character shrilling in his first few appearances in Green Lantern. He was really redeemed by his role in JLA. The idea is to allow a character to develop naturally, which is a great deal harder if your character is filling someone else's shoes. The character must make mistakes and learn from them, rather than getting everything right the first time.
The idea of a female Bond, therefore, would depend upon the 'Bond is a codename' theory becoming fact. This would anger a lot of the purists, who would argue 'Bond' is always the same person. Changing his race would raise other issues. My way to tackle the problem would be to introduce a new female agent in one movie, then give her the starring role in the next.
Chris