Yes, it is imperative to get our dressage horses cross trained! Riding on different terrain, reasonably uneven footing, pavement, grass, hills, etc. all helps to strengthen the tendons and joints against damage. I also jump my dressage horses, and take them out to big places so they can run.
My coach likes to take the horses outside to school the short work, or do other things and for young horses has a mandatory once a week hill session.
For reference, I am a CDI competitor. Yes, there are lots of people who have horses that do not see outside, but I think the horses are far healthier and more willing to work when they are let to go outside and be horses. This idea is gaining lots of popularity and with riders like Ingrid Klimke really promoting it and showing good results with her training program, the wheels are turning.
They say money doesn't buy happiness -- well happiness doesn't buy horses!