Bloodgazer Swarm!

New WQ added in 7.1 as part of a "swarm" set that give Wardens rep. Full info about the pets and quests for each of the 4 new "swarm" quests here. Each WQ is a progress bar that grants progress depending on what you kill (see top of page for full breakdown - fastest is 7 [npc=115674] kills). The [npc=115675] tend to pull in a pack.

Further to the Wardens rep, if you have the right food and kill the [npc=115674], you will be able to get a new battle pet that offers a quest chain with various rewards.

Make sure to have [item=128843] in your bags or you won't be able to interact with the [npc=115741] to get your [npc=115787]. Closest vendor is [npc=109372].

/way Azsuna 35.57, 5.66 [npc=115674]
/way Azsuna 35.5, 7.5 [npc=115741]

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As of Patch 7.1 there are 4 new Falcosaur pets. Each of the pets is associated with a different world quest around the Broken Isles. These pets all provide a quest chain to obtain a different mount. These quest chains require a couple of weeks to complete as each subsequent quest becomes available from your summoned pet after the daily reset after completing the previous quest.

To get each pet, you must first kill the Matriarch in the world quest area, which will cause the NPC version of the pet to spawn. The next step is to go to the NPC and interact with it and feed it its favourite food. If you don't have the food, you won't obtain the pet. There are vendors selling the required foods at the closest flightpaths to each world quest for most of these so just remember to pick some up on your way there!

Important note on orphans: I've encountered issues with the orphans not spawning after killing the matriarch. This isn't affected by the matriarch's nest but it does seem to be affected by getting the last hit on the matriarch. If you can't see the orphan after killing the matriarch then you may need to wait for the respawn. To check if you have completed the requirements to see the orphan run the scripts below.

[npc=115784] - Snowfeather Swarm!
This pet can be picked up from [npc=115737] in Highmountain (/way Highmountain 32.5, 28.2) after killing the [npc=115673] (/way Highmountain 35.1, 21.9). The food it wants is [item=128839], which can be picked up from [npc=108504] (/way Highmountain 29.8, 40.0).
Summoning this pet offers you the quest Raising Your Snowfeather which starts the quest chain to obtain [item=137578].
Subsequent quests: The Smell of Humans (Alliance) or The Smell of Orcs (Horde) > Hunting Lesson: Northern Hawk Owls > Allies in Highmountain > Snowfeather Team Up > Deadly Prey > Hunting Lesson: Spiketail Beaver > The Unfriendly Faction > Snowfeather Bonding > Hunting Lesson: Mudshell Conch > Teamwork Lesson: Naraxas > Hunting Lesson: Coralback Fiddler > Teamwork Lesson: Ursoc > Snowfeather Team Rumble > Snowfeather Reunion
/script print(IsQuestFlaggedCompleted(44879))

[npc=115787] - Bloodgazer Swarm!
This pet can be picked up from [npc=115741] in Azsuna (/way Azsuna 35.5, 7.5) after killing the [npc=115674] (/way Azsuna 35.6, 5.7). The food it wants is [item=128843], which can be picked up from [npc=109372] (/way Azsuna 47.2, 41.4) - this one is a long way from the quest area so try to plan ahead or you'll be running a long way to find food and will have to kill the matriarch again.
Summoning this pet offers you the quest Raising Your Bloodgazer which starts the quest chain to obtain [item=137577].
Subsequent quests: The Smell of Draenei (Alliance) or The Smell of Blood Elves (Horde) > Hunting Lesson: Erudite Manafiend > Allies in Azsuna > Bloodgazer Team Up > Dangerous Prey > Hunting Lesson: Fledgling Kingfeather > The Unfavorable Faction > Bloodgazer Bonding > Hunting Lesson: Juvenile Scuttleback > Teamwork Lesson: Serpentrix > Hunting Lesson: Felspider > Teamwork Lesson: Skorpyron > Bloodgazer Team Rumble > Bloodgazer Reunion
/script print(IsQuestFlaggedCompleted(44882))

[npc=115785] - Direbeak Swarm!
This pet can be picked up from [npc=115742] in Stormheim (/way Stormheim 79.8, 69.6) - jump down from the rocks above to reach here - after killing [npc=115671] (/way Stormheim 79, 76). The food it wants is [item=128763], which can be picked up from [npc=98112] (/way Stormheim 71.6, 60.6).
Summoning this pet offers you the quest Raising Your Direbeak which starts the quest chain to obtain [item=137579].
Subsequent quests: The Smell of Dwarves (Alliance) or The Smell of Tauren (Horde) > Hunting Lesson: Rose Taipan > Allies in Stormheim > Direbeak Team Up > Dire Prey > Hunting Lesson: Tiny Apparition > The Unkindly Faction > Direbeak Bonding > Hunting Lesson: Golden Eaglet > Teamwork Lesson: Fenryr > Hunting Lesson: Stormstruck Beaver > Teamwork Lesson: Krosus > Direbeak Team Rumble > Direbeak Reunion
/script print(IsQuestFlaggedCompleted(44880))

[npc=115786] - Sharptalon Swarm!
This pet can be picked up from [npc=115740] in Val'Sharah (/way Val'Sharah 47.0, 10.5) after killing [npc=115672] (/way Val'Sharah 48.0, 9.8). This world quest is somewhat awkward to get to but I recommend coming in from Highmountain (glider from a high point in the north-west or water walking around from the coast). The food it wants is [item=128837], which can be picked up from [npc=96784] in Dalaran (/way Dalaran 52.6, 34.4) - there are a variety of vendors in Highmountain who also sell the item should you choose to get to the world quest from there.
Summoning this pet offers you the quest Raising Your Sharptalon which starts the quest chain to obtain [item=137580].
Subsequent quests: The Smell of Night Elves (Alliance) or The Smell of Undead (Horde) > Hunting Lesson: Spring Strider > Allies in Val'sharah > Sharptalon Team Up > Destructive Prey > Hunting Lesson: Terror Larva > The Uncongenial Faction > Sharptalon Bonding > Hunting Lesson: Auburn Ringtail > Teamwork Lesson: Dresaron > Hunting Lesson: Vale Flitter > Teamwork Lesson: Xavius > Sharptalon Team Rumble > Sharptalon Reunion
/script print(IsQuestFlaggedCompleted(44881))

Check out [guide=4788] by Purgatorywolf for an in depth guide which includes details on requirements and tips for completion of each quest for your mount.

The best of the posts on acquiring a new hatchling pet while doing any of the Falcosaur Swarm! World Quests:

• Read the excellent guide Falcosaur! by Purgatorywolf.
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• See the quick food list for hatchlings from Nerrevar.
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• To see food vendors, click on a link to the food and view the "Sold by" tab. Click on the heading "Location" to sort vendors by zone. (Faction info is provided.) Vendors in Dalaran sell all the foods -- but not at one location.
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• See the locations of the matriarchs and hatchling orphans in this comment by Vampyrr. The related World Quest is [Falcosaur type] Swarm!
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• Fight the matriarch and adds that spawn with a player of your faction, bring a bodyguard.
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• The tiny hatchling does not drop from the matriarch but appears in a separate nearby area away from mobs. Click to "feed" and it goes into your inventory.
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• Give the left over hatchling food to a nearby player who forgot (it's not consumed).