Great experiment. I have seen the same experiment done with balloons in a car. Put a balloon in the middle of your car and take off. Why do the balloons move forward instead of backwards like you would expect?

Without the vase, the flame wants to stay in the same place. But you are moving the candle (and, hence, the bottom of the flame), so only the top of the flame stays put (briefly), so it's pointing in the direction it came from.

With the vase, the air around the flame is part of what is moving, so the flame "moves" less (but really stays in its old place less) because the "same place" is moving more. But I don't understand why it moves forward instead of lagging behind as in the first part. It doesn't seem to be inertia at all.

My guess is since inertia is "the tendency to stay in one place unless moved upon by an outside force", what happened was this. In the first move ( without the vase.) the flame was moved by the air, but in the second example with the vase, the flame moved in the same direction because it was acted on by the outside force of the plate moving to the left and NOT by the air blowing it so it would make sense for it to move in the same direction as the plate/candle/etc.

Here's my guess. No native speaker, I hope it's understandable, though.

Hot air (the flame) is lighter than cold air (surrounding the flame).

Moving the candle w/o vase.
When the candle is moved, the air cold air surrounding the flame will not move the same velocity due to inertia. Hence, the flame drags behind.

Moving the candle w/ vase.
When the vase is moved, all the air inside gets moved with it -- one side of the vase pushes the air in the direction of the move. If we move from left to right, it's the left hand side of the vase. Because of inertia, the air inside the vase gets compressed -- but more compressed on the left and less compressed (less dense) on the right. Together with the flame being lighter than air, it gets pushed to the right.

Best wishes from Europe. :)

Here's my guess. No native speaker, I hope it's understandable, though.

Hot air (the flame) is lighter than cold air (surrounding the flame).

Moving the candle w/o vase.
When the candle is moved, the air cold air surrounding the flame will not move the same velocity due to inertia. Hence, the flame drags behind.

Moving the candle w/ vase.
When the vase is moved, all the air inside gets moved with it -- one side of the vase pushes the air in the direction of the move. If we move from left to right, it's the left hand side of the vase. Because of inertia, the air inside the vase gets compressed -- but more compressed on the left and less compressed (less dense) on the right. Together with the flame being lighter than air, it gets pushed to the right.

Best wishes from Europe. :)

Because of the wind, the flame goes in the opposite way of the movement.
With the glass on top, there is no wind. So, it has to go upwards. BUT because of the inertia, inside the glass, it's the air which goes in the opposite way of the movement and the flame in the opposite way of the air (air down:flame up - air goes left:flame goes right - air goes right:flame goes left - etc...) so the flame goes in the same way as the movement...!!!