<Gives text of original conversation with Brandon>
To restate the scenario in more understandable terms (phase 2 is to use diagrams, if it comes to it and I still don't manage to get it across):
Say Cory the cork-thrower is standing besides a train track. Cory is facing North and the train is running from West to East. Cory tosses a cork North up over a passing train. Normally, this cork would go over the train and land on the ground directly opposite Cory to the North.
From the frame of reference of Cory and his cork, the train is moving West->East. From the frame of reference of the train, the train isn't moving at all and the cork is moving both South->North and East->West (i.e., Northwest). So if we were to draw a line describing the cork's movement, Cory's line would have the cork moving South->North over a moving train. The train's line would have the cork travelling Southeast->Northwest as it described a diagonal across the train.
If there's a bubble on the train, that's where things get complicated. When the bubble hits the cork, does the train's frame of reference "take over" so far as it's direction of travel goes? So far as the train is concerned, nothing really changes: the cork is still describing that same diagonal, just more quickly/slowly. But so far as Cory and his cork are concerned, all the sudden the cork is moving laterally (East->West) corresponding to the train's frame of reference. The question, then, is where the cork lands when all's said and done: does it still land directly North of Cory after it passes over the train, or does it land a bit to the West or East as well?
My thoughts/model on this would be that it also lands West/East. If the bubble was a bendalloy bubble, then the corks diagonal passage would be accelerated, meaning that it pops out of the bubble off to the West of where it would have otherwise. A cadmium bubble would still move the cork to the West according to its frame of reference, but because of how slow the bubble itself is in motion the cork would still end up East of Cory.
The bubble's frame of reference would take over while it's inside. But you also need to include the fact that bubbles deflect things. The cork would be deflected both when it enters and when it leaves the bubble. So you can't completely predict the path it will take.
<At this point the conversation kept on for a bit as things grew... complicated. We misunderstood one another [which I take the blame for] on several crucial fronts and ended up talking past one another. Long story short is that I'd been implicitly assuming absolute relativity of reference frames in the cork-bubble system—so while both types of bubble would drag the cork along for a bit, that dragging would also be offset (to varying degrees based on bubble type/compression) by lateral movement of the cork within the bubble. This is wrong.>
If the train is moving east, and he throws the cork over the train, a bubble that slows the cork down will mean the cork ends up east of him.
If the train is moving east, and he throws the cork over the train, a bubble that speeds the cork up will mean the cork ends up on the other side of the train faster than it would have with no bubble. It doesn't move west.
If the speed bubble only very slightly increases the flow of time, then the cork could even end up slightly east of him, depending on the speed of the train.
So depending on the speed or slowness of the bubble, and the speed of the train, the cork will either end up exactly where the thrower expects it to, but more quickly, slightly east of where he expects, but more quickly, or quite a bit east of where he expects, more slowly. The cork doesn't move west.
In fact, I think it's safe to assume that the train is always moving to the east faster than the thrower is throwing the cork to the north. In that case, both types of bubbles will always end up pushing the cork at least somewhat to the east.
Let's do the math here.
Say the bubble is 10 feet in diameter and the cork toss hits the bubble right in the center. He tossed the cork at 5mph. The bubble is 2x speed. That means the cork goes 10 mph across the train (measuring from the frame of reference of the tosser). The train is moving at 50 mph. The cork crosses the train in 0.682 seconds. In that time the train moves 50 feet to the east. So the cork ends up 50 feet to the east of where the tosser expected it to.
If the bubble is 100x speed, the cork goes 500mph across the train, and in that time the train moves 1 foot. The cork ends up 1 foot to the east of where the tosser expected it to, but much faster than he expected.
If the bubble is 1/2 speed, then the cork goes 2.5 mph across the train. The cork crosses the train in 2.727 seconds. In that time the train goes 200 feet to the east. The cork ends up 200 feet to the east of where the tosser thought it would end up.
If the bubble is 1/100 speed, then the cork goes 0.05 mph across the train. The train moves 1.9 miles in the time it takes the cork to cross the train. The tosser has no idea where it ends up, but he watches it hovering over the train as the train goes off into the distance.
As far as the cork is concerned, it can't tell the difference whether it's moving through a stationary bubble or a (laterally) moving bubble. From the cork's point of view it moves in a straight line either way.
<Some doodles got involved at one point or another, and it was also confirmed that the path of the cork (barring refraction) would stay the same once it left the bubble, still going directly north>