With the coastal missions patients that are not treated and transported by coastal helicopter will be taken by a boat back to their dock / rescue station at this time you get a follow up mission (patient transport) you send ambulances to that mission and they treat and then you transport to hospital
Alternatively if the mission has 5 patients and only requires 1 coastal helicopter you could send 5 directly to the mission to treat and transport the patients
Prisoners will always be brought back on a boat to their dock / rescue station they then are dealt with in a transfer mission by sending police cars that can transport prisoners to a cell
As far as transporting patients, it seems to depend on the mission. Had one where they went to the base for further transport, and another that went directly to the hospital.