Battle Pet Tamers: Warlords

Account-wide weekly quest added in 6.2.2 to defeat 5 of the possible 7 Draenor tamers throughout the week (Tuesday-Monday):
(It looks like this quest has experienced some changes as of 7.0.3, researching as I'm able.)

• Progress is shared between characters as of 7.0.3.
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• A tamer can be fought multiple days within the same week and count for credit (tamer's quests reset daily).
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• Progress also does not currently reset on Tuesdays with other restarts.
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• 7/26/2016 (7.0.3) - So far it looks like if you abandon the quest and reaccept it, you maintain the progress you had. I abandoned at 2/5 and when I reaccepted, I was at 2/5 again.
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Cymre Brightblade in Gorgrond

Strategy here for team:
[npc=68659] (1,1,1) + [npc=64899] (1,2,2) + Carry/Any

Pets are Legendary and all 3 pets heal. Remember you only need to defeat 5 of the listed tamers.

Vesharr in Spires of Arak

Strategy here for team:
[npc=68662] (2,2,2) + [npc=64899] (1,2,2) + Carry/Any

(For the rest of the day, after defeating Vesharr, you can buy [item=120051] (10g) and/or [item=120050] (1000g).)

Tarr the Terrible in Nagrand

Strategy here for team:
"Any two level 25 grubs with [petability=364] and [petability=160]" + Carry/Any

(Maggots = Larva = Grub(bs) for this attack set, check your inventory)

Taralune in Talador

Strategy here for team:
[npc=68662] (2,2,2) + [npc=62395] (2,1,1) + Carry/Any

Pretty much "two dragonkin with magic abilities", see also: [npc=68845], [npc=54027], etc.

Gargra in Frostfire Ridge

Strategy here for team:
[npc=40295] (1,1,1) + [npc=55356] (1,1,2) + Carry/Any

Most mechanicals can finish this fight. Might want shield/plating/heal, depending on crits.

Ashlei in Shadowmoon Valley

Strategy here for team:
[npc=62395] (2,1,1) + Carry/Any + Carry/Any

Easiest 2-pet carry. Fight is still over quickly if you don't want/need to level anything.

[npc=90675] (Alliance) or
[npc=91026] (Horde)

Erris: Critters of Draenor, pets can vary day-to-day, list here.
Kura: Critters of Draenor, pets can vary day-to-day, list here.

These tamers will have one of five possible teams. Wowhead will have the team of the day listed on their homepage under "Today in Draenor" on the sidebar.

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TIPS

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• Defeating all 6 Grand Master Tamers will grant Taming Draenor. Grand Master Tamers are fought once per day per account.
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• The character-level dailies at your menagerie from [npc=79858] (Horde) & [npc=85418] (Alliance) do not count (they aren't listed, either, but in case anyone was still wondering).
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TomTom (245/255):

/way Shadowmoon Valley:Draenor 50.0 31.2 Ashlei
/way Gorgrond 51.0 70.6 Cymre Brightblade
/way Frostfire Ridge 68.6 64.6 Gargra
/way Talador 49.0 80.4 Taralune
/way Nagrand:Draenor 56.2 9.8 Tarr the Terrible
/way Spires of Arak 46.2 45.4 Vesharr

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I picked up this quest from Lio, then went pet battling around Draenor, noting with satisfaction as the count rose with each Master I beat. I hearthed back to my garrison, mentally calculating how I would use my stones. As I exited Town Hall and headed to my menagerie, I noticed that Lio had a yellow exclamation over her head. When I clicked and realized she was offering me the quest I thought I just completed, my eyes frantically scanned my quest log, then my inventory. No quest. No stones.

Somewhere between finishing beating my last pet battlemaster and hearthing home, the quest vanished!

A weird bug just happened to me: I picked this quest with my main and beat Erris, progress 1/5. Then, I logged out and entered with an alter. After completing the garrison missions I had in queue, the progress updated to 5/5 and ready to turn in. Weird...

A weirder bug: I just opened a [item=140590] and my quest progress went from 0/5 to 5/5. This quest is broken as heck!