How can I safely transport cooked food in warm weather?
With all cooked, perishable food the total time in the so-called "danger zone" 40-140 °F / 4-60 °C must be minimized to avoid growing bacteria and thus food-borne illnesses. (Your transporting plans place your dishes flatly in the danger zone, so not a good idea.)

If your kitchen is on site, this is typically done by keeping food hot between cooking and serving. With many dishes this can be done for a short time only (a few hours maximum) before quality decreases.  

If cooking off-site on a comercial scale, there is often the "cook and chill" process used. 
In short, food is cooked, then cooled down rapidly to refrigerator temperature. It is transported cold to the cafeteria and reheated there in special ovens, hot water bath or simply on a stove.

Prepared food can be stored for up to five days, but you need special cooling/heating equipment and cold storage/vans.
(A manufacturer's explanation here.)