The Lost Dog can first be found in Pandaria’s Valley of the Four Winds:

/way Valley of the Four Winds 42.4 50

Pandaria - Sunsong Ranch

At 12600/21000 reputation with The Tillers you will be able to pick up Lost and Lonely. Completing it will make Dog join you at the Sunsong Ranch.

Draenor - Garrison

After moving to Draenor, Dog follows us and sits near our Herb Garden in the Garrison.

Legion - Pet Menagerie

After Legion launched, 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 Loose Pebbles which are very difficult to see:

/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 Pebbles, hearthstone back to your Garrison and interact with Dog, and then he will show up in the Pet Menagerie in Dalaran.