I think there might have been a discussion about this in the comments to some other question a few months ago… somewhere. Not sure I remember it all that clearly now. Hmm.
The eventual conclusion was that stronger Pokéballs are more effective because they are more comfortable for the Pokémon inside them. I think that being in a Pokéball is a lot like dreaming; their awareness of the world around them is dimmed and their perception of time goes completely out the window, but they can still hear, and they can ‘wake up’ if they realise something is seriously wrong. Great Balls and Ultra Balls might introduce a mild euphoria into this sensation – a pleasant dream, if you will – so that the Pokémon simply enjoys being inside enough to think twice about a trainer it might otherwise reject. After all, the trainer has gone to the trouble of buying a more expensive Pokéball, or selecting a more appropriate one.
That does still leave a major difficulty, though: dealing with all the specialised types of Pokéball. Some are reasonably easy to explain – I imagine that being in a Lure Ball feels like dreaming of swimming, while perhaps being in a Moon Ball feels like being bathed in moonlight. Others, not so much – some Pokéballs work based on circumstance, like Dusk Balls and Timer Balls, and I have no idea how those would function. The most troublesome is the Luxury Ball, since the name seems to imply that it is extremely comfortable for Pokémon, and makes them more friendly as a result, but has no effect on how easy they are to capture in the first place.
In short, this is my best guess at how they work, but I am painfully aware that some types of Pokéball simply do not appear to fit the model.
Pokémaniacal 