Kids have always enjoyed reading about the same characters in different stories. When I was young I loved the Heidi trilogy, Little Women, Nancy Drew, the Ramona books, the Narnia series and many more.
Some are justifiably classics while others weren't very well written, but kids still check them out of the library or get them as gifts and enjoy them.
Recently I read one from a series that I won't mention by name because it was so bad. The family of children told their parents they were about to take their boat to sea to follow the evil bad guys and the parents response was to suggest they try to be home in time for dinner. Even in the 1920s or 30s when the book was written that was unrealistic.
But some recently written series are annoying in a different way.
I find it very frustrating when I come to the last page and nothing has been resolved. That's just a dishonest way to get people to buy the next book, and then the one after that, and so on because they'll never find out what happens unless they read the entire series. It's okay if there are still a few unanswered questions at the end of a book, but the main ones should be resolved or the reader has been cheated.
Of course all newer series for kids aren't like that.
Some great ones I've been reading lately are the Enola Holmes Mysteries by Nancy Springer.
Each book has a mystery that is solved by Enola, the younger sister of Sherlock Holmes, at the end, but there's an underlying problem of the girl's family relationships that continues throughout the series.
Springer does a wonderful job of portraying the world of long-ago London, (some of which might be disturbing to sensitive young readers) and the historical setting is essential to the plots, not just an interesting background. The feisty protagonist is someone modern girls can relate to, the stories are exciting, and the mysteries difficult for the readers to solve.
I highly recommend this series for kids.