I have just watched "The Secret Horse: Quest for the True Appaloosa" on the BBC. I hadn't expected that they would end up in Kyrgyzstan. The valley where they ended up was pretty isolated and the locals said they had resisted the importation of Russian horses under the Soviets.
Whether there really is a link between US Appaloosa and the Kyrgyz Chaar horses, this gives those who like a bit of variety in their horses the excuse to paint some as Appaloosas. This varies from mottled hind quarters up to all over Dalmatian style spots.
Only available on Iplayer until 20th Feb though there are some details and further links here http://www.historyextra.com/feature/animals/how-did-appaloosa-horse-get-north-america
Conventional wisdom has it that horses did not (re-)arrive in the New World before the Spanish, none of the natives knowing what these strange new 'deer' were. But from how wide a background would the Spanish have obtained their horses? I think we can speculate on a link Kyrghyz-Turks-Balkans or North Africa-intermediate horse traders-Spain, possibly even via Stradiots at some point. All speculation, of course, but as Peter hints, speculation perhaps worthy of a paintbrush.
"The Spanish probably obtained spotted horses through trade with southern Austria and Hungary, where the color pattern was known to exist", says Wikipedia.
And then again, "Today, leopard complex spotting phenotypes are frequently found in a range of breeds from Asia (Altai Horses, Mongolian Pony), America (Appaloosa, American Miniature Horse, Colorado Ranger Horse, Falabella, Pony of the Americas, Spanish Mustang) and Europe (British Spotted Pony, Knabstrupper, Noriker, Karabaier). Leopard complex spotting (LP) is inherited in an autosomal incomplete dominant mode." - http://rstb.royalsocietypublishing.org/content/370/1660/20130386