When should the greeting of young girls stop and one turn away from them, and at what age?

The madhahib (schools of thought) agree on the impermissibility of looking at or shaking hands with a young girl if it is accompanied by lust. In the absence of lust:

Malikis and the sounder opinion among Shafi'is: It is impermissible to look at or shake hands with her if she has reached an age where she is typically desired. Hanafis and Hanbalis (the relied-upon view): It is impermissible to look at or shake hands with her after the age of ten, as her 'awrah (intimate parts) becomes like that of a free, adult woman. Shaykh Ibn 'Uthaymeen: She takes the ruling of an adult woman, and modesty becomes obligatory upon her if she reaches an age where men's souls become attached to her, even if she has not yet reached ten. This varies depending on the girl's growth.