Is it permissible to take the daughter of a maternal aunt and raise her away from her mother, because her mother entered into an 'urfi (customary) marriage without her family's knowledge, in addition to her having other relationships?

An adult girl is not subject to custody. As for a girl who has not reached adulthood, if she reaches nine years of age, scholars have differed regarding who is entitled to her custody. The questioner has no right to the custody of her cousin's daughter when there is someone closer than her, such as the aunt. If a dispute arises, the Sharia court is the reference for resolving it.