The https://www.wowhead.com/npc=59533/lost-dog can first be found in Pandaria’s Valley of the Four Winds:
/way Valley of the Four Winds 42.4, 50.0
At 12600/21000 reputation with The Tillers you will be able to pick up https://www.wowhead.com/quest=30526/lost-and-lonely, which upon completion will make https://www.wowhead.com/npc=59536/ join you at the Sunsong Ranch.
After moving to Draenor, https://www.wowhead.com/npc=87553/dog follows us and sits near our Herb Garden in the Garrison.
After Legion launched, https://www.wowhead.com/npc=87553/dog was left alone in the Garrison, until the secret was solved on how to transport him to new Dalaran. Namely, scattered throughout Dalaran there are 10 https://www.wowhead.com/object=269037/loose-pebbles which are very difficult to see: https://i.imgur.com/Olw3LeV.jpg
/way 46.15, 53.60
/way 40.40, 53.50
/way 37.47, 28.77
/way 43.99, 17.47
/way 49.15, 70.45
/way 47.55, 28.69
/way 42.73, 43.73
/way 53.92, 41.25
/way 53.30, 52.33
/way 52.01, 61.98
Loot one of these https://www.wowhead.com/item=147420/pebbles, hearthstone back to your Garrison and interact with Dog, and then he will show up in the Pet Menagerie in Dalaran.