This idea will work well with either propane or natural gas grills, not so well with charcoal grills.
Make the cooking chamber completely air-tight, and channel all of the exhaust gas through a counter current entropy exchanger (a heat exchanger that's steam permeable).
Meanwhile, make sure that
all of the fresh air that the grill will use (before it's mixed with the fuel) passes through this same entropy exchanger (EX).
As the cooking gas burns, it generates H20 as a combustion product, which increases the humidity of the air in the grill. When this air goes out the EX, the moisture transfers over to the incoming fresh air.
Eventually, the relative humidity of the air in the grill will be close to 100%. As a consequence, almost no moisture from the food will be able to evaporate, and drying of the food will be reduced. Also, evaporation absorbs heat, so if evaporation is reduced, heat loss is reduced.
Furthermore, since the food will continue to absorb heat for as long as the food's temperature is below the grill's air temperature, some steam may actually condense onto the food. If there's a smoker box in the grill, then that condensate will be very smoke-flavored.
Another consequence of the steam in the grill's air will be increased thermal conductivity, which will result in faster cooking.
And of course, any transfer of heat from exhaust to intake will reduce the amount of fuel needed to heat the incoming air to the target temperature... thus increasing the grill's fuel efficiency.