The puppy behaves as a puppy would behave. He is hyper and quite active. His temperament and disposition matches the narrator's propensity for being innocently mischievous. The puppy trashes the mother's garden, bothers her, creates a mess around the house, and is fairly disruptive. The puppy is active, teeming with life. Given the tension that the mother is displaying because of the family's move, the puppy is not helping issues and receives much of the mother's wrath. The puppy is who ends up finding the two men, hiding in the garden and threatening both himself and the narrator.
In the end, the puppy is left behind. Liberty's loyalty to the narrator is strong enough that the narrator has to kick him to get him moving. He goes through the gates into an unknown setting, "down the drive, through the big gates, to the world out there." Liberty ends up being loyal and honorable in a time and setting where honor and loyalty were not necessarily valued. In this condition, he is left behind wth the hope that the narrator can find her "liberty" later on in life.