I think it's your computer remembering what the last page is. The horses, I doubt are moving. The computer is showing you the last thing it remembers on that page.
What browser do you use? It happened to me when I started using Opera. I had to change the cache refresh speed to automatic, because before it would refresh it and get new information (as in, realize that I moved a horse or bought an item) only every couple minutes, which wasn't fast enough. If that's your problem I can tell you how to do it.
I've had this problem too. The only thing is, the horses that are showing up aren't the ones I had in the stable that day. (I.e, confy horses in stable when I've been working with the sports)
45mil still needed...
And what rough beast, its hour come round at last, Slouches towards Bethlehem to be born?