Can we trade some efficiency to curb high-altitude emissions? Considering our problematic issues,


Emissions at high altitude
Contrails 
~10,000 premature deaths per year from airline emissions


This is an area of concern where it's worth asking some stupid questions.

What would happen if a hose was attached to the core/hot nozzle of a turbofan, diverted to storage on the body? Could it still be efficient 'enough' (assuming a highest possible bypass ratio) to compress and carry some of that carbon dioxide to the next airport?
There's a fundamental problem. A jet engine performs work by expansion of the combustion products. Compressing the combustion products would require at least as much work as was provided by the expansion of the combustion products. Only an external combustion engine (for example, a steam engine) would work in this case.

A second problem is that the mass flow of the CO2 is more than three times higher than the mass flow of the fuel. Your plane would get heavier as you fly, and since the max landing weight of a typical plane is lower than the max takeoff weight, this will severely limit the aircraft range.

All in all, you can probably manage to make a purpose-built plane that fits the bill but I highly doubt it will have much usable payload if any at all.