According to Wikipedia's entry for Blondie (the comic strip), of the 28 movies made based on the strip, at least the first 14 were a continuous series meeting your definition:
Blondie was adapted into a long-running series of 28 low-budget theatrical B-features, produced by Columbia Pictures. [...]
Columbia was careful to maintain continuity, so each picture progressed from where the last one left off. Thus the Bumstead children grew from toddlers to young adults onscreen. [...]
In 1943 Columbia felt the series was slipping, and ended the string with It's a Great Life and Footlight Glamour, deliberately omitting "Blondie" from the titles to attract unwary moviegoers. After 14 Blondies, stars Singleton and Lake moved on to other productions. During their absence from the screen, Columbia heard from many exhibitors and fans who wanted the Blondies back. The studio reactivated the series, which ran another 14 films until discontinued permanently in 1950.
So this is definitely 14, and possibly as many asmost likely all 28; I note that the entry for Larry Sims (the child actor who plays Dagwood & Blondie's son Alexander) is listed as appearing in all 28 films along with the two main stars, Penny Singleton and Arthur Lake.