More than one swollen lymph nose under one arm that moves. what could be causing this?

Regional disease: The most common cause of armpit (axillary) lymph node enlargement in a young individual is the presence of inflammation/infection somewhere on that side of the arm or chest. A less common cause is a malignant tumor (cancer) in that region, including the breast. Without assuming the worse, schedule a thorough exam with your primary care provider.

Lymph node : Lymph nodes can be a normal reactive process to infection or can be the result of serious diseases. Size and density are important. See a doctor