I agree that a clarification with regard to what is meant by "sport" would be helpful. I'm thinking more along the lines of foxes that are used for fox hunts, bulls used for bullfighting, or even the bulls used for the Running of the Bulls in Pamplona. These are definitely examples of how animals should not be used for sport.
I think that when we start having people breed animals only for the sake of participating in an activity that is only for the purpose of entertaining others we run the risk of causing all kinds of issues for the animals. I think about the greyhound rescue groups to save the greyhounds who are no longer able to race. I have even heard of breeders destroying animals that are born with what they consider less than desirable traits.
This depends on what you mean by "sport." I will give you a couple of examples that could be used to argue that using animals for sport is okay.
If you are talking about hunting, for example, then a person could argue that hunting is actually no less humane than other things that could happen to an animal. In some areas, there could be overpopulation by deer and many of them could starve to death. In other places, they could be killed by predators. Neither of these is better than letting people shoot them.
If you are talking about horse racing and that sort of thing, you could compare it to some sports people play. You could say that we allow people to endanger themselves boxing or playing football for our entertainment so what's so bad about endangering some animals for our entertainment?
1. Many animals actually thrive in a "working" environment and since many cannot live on a farm where they do have specific work, being involved in sport can serve that role for them providing all the benefits of it.
2. For many dogs in particular, learning new skills helps them to be more confident but also to understand their role as a submissive animal better. Sports and competitions are a great place for them to do this.
3. Sporting animals can be treated well and be empowered by sport.
1. You could probably split this into several points but the incredible cruelty practiced on animals in many sports, particularly racing sports is horrific.
2. When any kind of animal suffering is made into a public spectacle (bull fighting, etc.) it is an indication of a sick society.
3. The huge number of animals bred for racing that don't pan out and are killed or kept in horrible conditions is also disturbing.
In a world assaulted by engines, noise and the lack of respect towards nature and everything beautiful and healthy, I think that there is no crime to ride a horse in a competition, than driving a rally car. If the equitation is practiced in the manner that the rider to understand that a horse is not a car, but a gentle creature to be respected, not only an hardworked object, then it is ok to allow use of horses in equitation.
The key of any sport where animals are present is to treat that animal as a being and not an object. As long as the animal is your team mate, everything is just fine.