do you see the world slowly moving in the direction of phasing out the raising of animals for either food or bloodsport, over the course of centuries or generations?
Maybe, but not for the obvious reasons.
One might argue that there is a progressive trend in commonly shared attitudes towards animal welfare in the western world and one might extend this trend until a time where eating meat derived from animals would become taboo.
One might equally argue that in the West we have a hypocritical attitude, or at least one of willing self-deception towards animal welfare - people probably have an idea that their consumption of meat likely causes a level of animal suffering that they would find totally unacceptable for, say, their own dog or cat - and this hypocrisy/deception/disconnect could well carry on for some time (perhaps progressively more humane methods of farming and slaughter might also be adopted).
There's also the rapidly developing non-western economic powers - I might be displaying my ignorance of other cultures again, but the impression I get is that very few Chinese have much understanding of, or sympathy with, Western attitudes towards animal welfare. Perhaps this will change with time.
All that is by-the-bye, any attitudinal changes would probably require the generations or centuries that you mention and in that time I think it likely that human civilisation will undergo some profound changes.
As per the end of the world thread, I'm pretty pesimistic about our chances in the next 100 years.
IMO, we're not going to get to grips with atmospheric CO2 in enough time to prevent serious global warming. We've also got an impending energy, over-population and resource crisis. Added together, by the end of the century people may become vegetarian out of necessity - raising animals for meat will just be far too inefficient a use of scarce resouces (plus, the oceans will likely be fished out, and dying due to acidification).
OTOH, if we dodge the climate and energy/pop/resource crunch bullets then
the future is likely to be very strange with humans augmenting their brains using artificial intelligence and eventually self evolving into hyper intelligent beings. It's hard to predict what would happen from then on, but I would guess that such beings would ditch the cumbersome, inefficient biological system and use a direct energy source - thus vegetarianism goes out of the window by default.
Either, or, perhaps, some combination of the above pessimistic or strange scenarios will likely happen by the end of this century.
EDIT: I forgot about
this article in the Times yesterday