Yes, they are actually from the herd of the Falabella family, thus they were dubbed with that name to show where they originated. Chianti was one of the imports from Senor Falabellas herd, as were quite a few others. Falabellas also come in solids and pintos too, but to most, when the name is mentioned, loud Appy coloring comes to mind.
One of Senor Falabellas daughters (I believe) had moved to the U.S. and continued to breed, but Laurie Stevens at Toyland Ranch purchased the rest of her herd a few short years back, to continue with the tradition of the little horses that the Falabella family were so careful about breeding. If you contact Laurie, she is very nice and will probably be glad to answer any questions.