Remote, Emergency, Medical, Online Training, Telehealth, and EMT Act or the REMOTE Act This bill requires U.S. Customs and Border Protection (CBP) to take actions to improve CBP medical services along the southern border. CBP shall make emergency medical technician (EMT) and paramedic training available to selected border patrol agents along the southern border. Agents that complete the training or already have EMT or paramedic certifications shall receive specified pay increases. The bill imposes requirements related to (1) the percentage of agents with EMT or paramedic training assigned in each sector, and (2) minimum levels of various medical supplies in each sector. Furthermore, CBP shall require all border patrol agents and officers to complete online training on how to (1) identify common signs of medical distress in children, and (2) timely transfer sick children to medical care. CBP shall also ensure that all remote stations, forward operating bases, and remote ports of entry on the southern border have voice access to a medical command physician.