Milia: These are small, white or flesh-colored bumps that appear on the skin of many infants. They occur when dead skin cells or keratin, a protein, get trapped under the skin surface. Given the description of discrete, smooth, flesh-colored, dome-shaped papules, milia are a common and likely diagnosis in infants under 12 months.

Syringomas: These are benign growths that can appear as small, flesh-colored or yellowish bumps on the skin, often around the eyes but can occur elsewhere. They are more common in females and can start appearing in infancy.
Eccrine Poroma: A rare benign adnexal tumor that can present as a small, dome-shaped papule. Though less common, it's a consideration in the differential for flesh-colored papules.
Infantile Acrochordons (Skin Tags): While more common in adults, skin tags can occasionally be found in infants. They are small, usually flesh-colored, and hang from the skin.

Congenital Melanocytic Nevi: Although these are typically pigmented, some may be amelanotic (without pigment) and could present as flesh-colored papules. Early detection is crucial due to the potential for malignancy.
Leukemia Cutis: Cutaneous manifestations of leukemia can be varied and, in rare cases, might present as flesh-colored papules. This diagnosis is critical to consider due to its severe implications.

Trichoepithelioma: A rare, benign tumor of the hair follicle that can present as a small, flesh-colored papule. It's part of a spectrum of basal cell nevus syndrome in some cases.
Accessory Tragus: A small, usually flesh-colored, pedunculated lesion that represents an embryological remnant. It's typically found on the jawline or preauricular area.
Nevus Sebaceus of Jadassohn: A congenital skin lesion that can appear as a yellowish or flesh-colored plaque or papule, often on the scalp. It has the potential for malignant transformation later in life.