After acne lesions heal, brown spots can appear in their place, especially on the skins of people of color. These spots are caused by excessive melanin production and occur after an inflammation, like an acne outbreak or rash. Dr. Baumann recommends using a moisturizer that contains soy or niacinamide (a vitamin B complex), to help prevent the dark spots in the first place. People with darker complexions people may also experience the white, "ashy" appearance of dry skin, which can be relieved easily by using a lotion or moisturizer. As always, make sure that you choose a moisturizer that's non-comedogenic or non-acnegenic.
Otherwise, people of all skin colors have the same types of issues with moisturizing acne-prone skin and can benefit from reading ingredient labels and avoiding products containing oil.