I currently have my US Visa in an old expired passport and a new passport. I want to renew my passport and then my US Visa shortly after, so that the visa is in the new passport and their new expiration dates become close. I have a few years left both in the visa and the passport.
While from my research there's no official restriction in doing so, are there any drawbacks I should keep in mind? For example, will this "raise a flag" in the visa renewal interview or at immigration?