Inertia problem? Inertia of air?

Inertia as studied in physics or mechanics is resistance to motion.

I observed a fly and a butterfly travelling in a bus. Bus had open windows and open door, so air was freely coming in and going out. As the bus accelerated or decelerated, I experienced jerk or inertia. However, the fly was hovering in front of me in the same place. Can you explain this phenomenon with regards to inertia to me?

I expected the fly to move back (with respect to me) as the bus moved forward, but it didn’t happen. I thought inertia of air should be much less than mine to let it happen.

What would have happened if the bus had all windows and the door closed? And, what would happen if the fly or butterfly would enter in airplane or helicopter?