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I don't think that there would have been much direct contact in latter days between Gondor and Rivendell. Faramir had clearly never seen an elf when he met Frodo and only Denethor had much of a clue where to find Imladris.
If there were any Elf/Gondor interaction my guess would be by ship between Dol Amroth and Lindon. I think the Elves would have had a tradition of horse breeding but I guess that Gondor would have traded extensively with Rohan in later days since they didn't have a strong tradition of using riding horses - I think UT refers to them lacking enough horses capable of carrying men of Numenorean stature.
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