The "Little Women" series is foremost a bildungsroman, literally 'education novel' from the German (where it originated, in the first half of the 18th century), meaning a 'coming of age' narrative in which the protagonist(s) mature in either or both age, wisdom and/or morality. Morality is a very significant part of this genre and the progression of the protagonist(s) to (a) moral and wise character(s) is depicted as a key part to the narrative. This can be exemplified in Jo's heartbreaking rejection of Laurie's marriage proposal, in which she shows wisdom far beyond her years as she recognises that Laurie's fierce passion for her is not a good basis for a marriage. Instead she eventually marries Professor Bhaer, an older but a much more sensible husband than the charming but impulsive Laurie. This is a stark contrast to Joe's characteristic of being somewhat childish (in a good way!) behaviour seen earlier in the novel, such as her over-enthusiastic renditions of plays. She also metamorphoses from being rebellious, outspoken and often angry to a profoundly psychologically mature woman.
This psychological progression of Jo's is characteristic of the bildungsroman genre. Dickens is another novelist who widely uses this technique, especially in Great Expectations and Oliver Twist. In terms of L.M Alcott's novels being a backlash to immoral works of the period, this could be referring to the boom in Gothic or Sensation Novels in the 1850s-60s which were written to achieve the goal of shocking and scaring readers. Charles Darwin's The Origin of Species was also written in this period which was and still is a controversial work. Her novels remained innocent, as a silent protest to her contemporaries.
For more about the bildungsroman genre, this site explains it well.