Homegrown manufacturing, shipped via a lengthy trans-Atlantic supply line to various European ports, a distribution point in Geneva, and on to various camps.

If you're interested in this system, I recommend Confronting Captivity: Britain and the United States and Their POWs in Nazi Germany. Despite the awkward subtitle, it's a reasonably complete and modern work from the University of North Carolina Press and goes right to your question.

At the outbreak of the war, the International Committee of the Red Cross, and its British chapters in particular, were unprepared for the magnitude of the demand they would face. Manufacturing the parcels was slow, stockpiles were low, and the logistic chain was inadequate. This problem was exacerbated after the fall of France. Instead of shipping parcels across the Channel, the British set up arrangements to send parcels to Lisbon, across Spain and unoccupied France, then on to Geneva and camps in the German territories.

The Germans and the British both were initially unprepared for the vast numbers of POWs created by the fall of France. Letters from imprisoned British soldiers spoke of harsh conditions, cruel guards, and a lack of Red Cross support. The failure of the Red Cross parcel system led to wholesale changes in its administration during the latter half of 1940 and the first part of 1941.

When the United States entered the war, it inherited the British model of doing things, then improved upon it. Packing centers were set up in New York, Philadelphia, St. Louis and Chicago, according to the Red Cross history of the effort. More than 13,500 volunteers assembled the packages, which ─ according to the Smithsonian Museum of American History ─ "sually contained the following: Army spread, canned bacon, luncheon meat, salmon, dehydrated corned beef, canned cheese product, soluble coffee, powdered milk and chocolate D bars. Non-food items included: toilet paper, soap, paper towels, can opener, needles, thread, patching cloth, vitamin capsules, salt and tobacco."

There were special packages for Christmas, for the disabled, and for other special needs as well.

Special Red Cross ships, chartered specifically for the Red Cross and carrying only these parcels and other relief supplies, sailed the Atlantic under the rules normally applied to hospital ships. Again, according to the brief Red Cross history of the effort, "These ships sailed singly – without convoy – along a designated route, with lights blazing at night upon their Red Cross markings."

Lisbon wasn't the only port where packages were landed. Barcelona was considered, as was Marseille and even ports in Germany itself. Hamburg is mentioned in some accounts, though this practice ended in 1943 with the firebombing of that city. After the invasion of Normandy, the flow of parcels became much more sporadic, as bombing of the German rail and road network disrupted the movement of parcels as well as other items.

When the war ended, similar systems were put into use on a much wider scale to feed the millions of refugees and Axis POWs left after the fighting.

As former POW Maynard Unger later said:

"If it weren’t for the food packages from the Red Cross, a lot of people wouldn’t have made it. I remember one particular time we went to pick up our Red Cross parcels. Believe me when I say these were very special occasions. They were important!