Not long after Aragorn was crowned King, he desired to speak to Ornessar, and so he did. He asked of him one favour; that the House of Ornessë would gather its forces one last time and assault the Angmarim that regrouped and took control of Carn Dûm, after the fall of Mordirith. Ornessar swore to fulfill the King's command, and he started his journey to Lothlórien the same day. Long and tiring was the journey, but he and Amdir, the horse that was given to him, made their way to Lórien relatively quickly.
When he was about to finally reach his destination, some Galadhrim scouts saw him from afar and they let the whole House of Ornessë (which was summoned there by him, to aid in the War in the North) know that he was coming. When he reached the gate of Caras Galadhon and just after he dismounted from his horse, they encircled him and hugged him. Ornessar's heart was beating very fast, for he was very joyous and excited that he met his family after so long; especially his beloved, Calaerneth. They talked for a while, and then they went to rest because tomorrow they would start making their way to Eriador.
And so they did, but of course, nothing significant happened during the trip. Middle-Earth was almost free of trouble now. Once they reached Imladris, they gathered anything they would need for their mission, they mounted on their war-horses, and they met near the Spire of Meeting, but that is another story.
But you might be wondering; what about Katrandil, the Forngarmo? Is she in Gondor? In Rohan? In Eriador? Did she enter Mordor with a small part of the Host of the West to destroy all that's left of Sauron's strongholds, in the Fourth Age? Or is she dead? The answer is simple - she would be in the service of King Elessar, and would eventually step into Mordor. But that's a story for another time.