You can go on the website of your federation, and to search with the name of the horse that you looking for. Normally in all countries, there is a database listing all the horses (I know the one in France "national studs"), I think that is where you will find the most information !!
I hope this helps :)
We have something like that organized by an equestrian magazine here in Spain. But they just upload the results of the horses and the riders in some competitions. Also this magazine has a website for stallions with their origins and best results…
I think ANCCE (National Purebred Spanish Horse Breeder's Association) and other associations have a record of all the horses registered with their origins and their performance.