Yes, but it takes some attention and effort.
The FEI launched a report about sustainability at equestrian events. You can read more about that here: http://thesustainablehorse.com/fei-launches-sustainability-handbook-for-equestrian-events/
The Swedish dressage tour Lovsta Future Challenge compensates the CO2 from the travels from all participants and judges to and from the shows. http://thesustainablehorse.com/swedish-lovsta-future-challenge-compensates-co2-emission/
It would be good if big tours as the Global Champions Tour also did something like this.
The European Championships 2017 in Goteborg will be fully sustainable, but also during the World Equestrian Games in France there were already a lot of sustainability issues covered.