Irritant Contact Dermatitis: This condition is a common reaction to environmental factors such as harsh soaps, detergents, or extreme weather conditions, which the patient may have been exposed to after moving to the US. The symptoms of dry skin, hyperpigmentation, and small bumps without pus are consistent with this diagnosis.

Seborrheic Dermatitis: Characterized by a red, itchy, and flaky rash, often on the face, this condition could be exacerbated by environmental changes and stress related to moving.
Eczema (Atopic Dermatitis): Given the patient's symptoms of dry skin and small bumps, eczema is a possible diagnosis, especially if the patient has a history of atopic diseases.
Keratosis Pilaris: This condition causes small, rough bumps on the skin and can be associated with dry skin, fitting some of the patient's symptoms.

Lupus: Systemic lupus erythematosus can cause a variety of skin manifestations, including a malar rash (butterfly-shaped rash across the cheeks), which might initially present with dryness and hyperpigmentation. Although less common, missing this diagnosis could have significant consequences.
Sarcoidosis: This condition can cause skin lesions and bumps, along with other systemic symptoms. It's crucial to consider due to its potential for serious complications if left untreated.

Dermatomyositis: An inflammatory disease characterized by skin rash and muscle weakness, which could initially present with skin manifestations similar to those described.
Granuloma Faciale: A rare skin condition that causes reddish-brown bumps on the face, which might be considered if other diagnoses are ruled out and the condition persists or worsens.