Run over to the Aphelion's Rest lost sector, and head inside. If you take a hit of queensfoil right as you run into the first bundle of thrall, you won't be able to miss it as you go through the lost sector.
As soon as you jump in, immediately start running, there's some knights that'll fuck you up. The only goal in this challenge is to kill all 4 blights (taken ball things) around the map, and then quickly get back to the middle and destroy all the purple crystals.
#2: Forfeit Shrine
Head over to the Gardens of Esila off of "The Strand" area. Once you start fending off enemies, chug a tincture, and run along the cliff to your left, going around large rocks as you need to. The entrance to the challenge is floating off the edge of the cliff right after you go around one of the rocks.
This one is another "run really fast" one. You're going to want to grab all 3 arc charges strewn about the map, and bring them to the very middle of the map, near where you spawn. There should be a glowy little light in the middle, that's where you know to dunk them. As you're running up to the dunk zone with a charge, taken boop mans will spawn to try and play defense. Just run 180 degrees around them and dunk behind them.
After you dunk all the balls, the light in the middle will turn into a little golden tree/shrub/plant thing. Running through it recharges your super within seconds, and starts health regen to boot. Your job is now to kill all the knights. You can do this however you want, whether you engage in super cool combat with supers flying back and forth as the knights fall one by one, or you hide in the tree nearby and just use your whisper. You do you.
#3: Shattered Ruins
For this one, you're going to be running basically all the way back to the place you spawned in when you first entered the dreaming city (if you can't remember back that far, you run almost to the end of the Spine of Keres area (same section of the map where you talk to the oracle). Keep going through the area until you get to the pillars of rocks you parkour between over a misty ground. Chug a tincture and you should hopefully see the portal nearby.
Basically parkour up to the very top of the area (dodging taken boops, taken bombs, or both). Once you get to the very top, there'll be a big yellow health boi that spawns. Kill him and you're good! If you can't find the exit portal, look up.
#4: Keep of Honed Edges
Head to the Harbinger's Seclude, the building off of the Rheasilvia zone. Head to the very back of the building to the huge statue. Take a swig of space-drugs and look to your right, you should see platforms that help you get onto the statue's head. You can also just climb up the statue itself (warlocks can do this ezpz), but who am I to tell you how to live your life.
This one's one of the longer ones. Start by parkouring over the floating rocks and dark dimension junk to get to the gazebo in the very middle that's so impossibly bright that I'm sure you're already half way there as you read this. Next, grab the sword in the middle of the gazebo, and use it to wipe out those knights in the middle. If you've got teammates, let them take care of the ads, that sword is the only thing that can hurt the knights. After you kill everything in the gazebo (why the hell is there a gazebo anyway), look around for the floaty white light that is toland, he should fly off in the direction you need to go. After this it's just more parkour until you get to the end, but watch out for giant flying rocks that fly in out of literally nowhere and kill you, making you have to start completely over. Those are fun.
#5: Agonarch Abyss
Fly into the dreaming city, turn around, go into the Bay of Drowned Wishes lost sector, take a swig of magic awoken liquid, and run past all the screebs trying to kill you to find the portal near the door that's blocked off by a purple force field.
You thought these all were ascendant challenges? Well think again fucker, this is a DESCENDANT challenge. Parkour down the fairly linear parkour route until you get to the gazebo at the end (again with the gazebos), kill the 3 witches in the gazebo, and bingo bango bongo you've done it.
#6: Cimmerian Garrison
Head to the Chamber of Starlight in the Rhaesilvia area and go through it until you get to the huge main room where the boss is. Drink the thing and then find the entrance to the challenge behind the loot chest, on the ledge with all the hobgoblins that are currently sniping you.
This one is probably the longest challenge out of all of them. Spawn in, turn around, and follow the really long path all the way through the map. It's really long, with plenty of ads spawning along the way. Occasionally you might have knights spawn right in front of or behind you, just jump out of their way before they two shot you. At some point the path will just become chunks of rock, just keep going until you reach the little grassy platform in the sky. Here, you just need to kill the knight on that platform. There are some pillars you can stand on, but be warned, there's shriekers who will shoot you down if you stay up there for more than a few seconds.
Nur Abath, Crest of Xol
This ogre son of a bitch is immune whenever any other enemy is near him. Clear the ads aroung him, and you can hurt him all you want.
Kathok, Roar of Xol
This big ass knight has a shield that can only be damaged by a sword that drops from Severing Knights that spawn during the boss fight. Kill the knight, grab the sword, break the shield, DPS the shit out of him, when his shield comes back up rinse and repeat.
Damkath, the Mask
This ugly ass ogre is immune all over his entire body except for a big, glowing, bulgey growth on his back. Shoot him there and you should be good.
Naksud, the Famine
Every IKELOS gun
This ogre looks weird as fuck, and has one of the hardest boss mechanics in Escalation Protocol. He's relatively squishy for a boss but after taking some damage some cursed thralls called "Naksud's Fodder" will spawn. If they can run all the way to the boss and explode on him, they will heal him FUCKING LOTS. Once you hear the hive spawn noise, everyone on your fireteam should immediately prioritize killing those thrall.
Bok Litur, Hunger of Xol
Every IKELOS gun
This knight is also a big ass knight, and he has a SHIT ton of health. Make sure you focus as much as you can on damaging him. He also occasionally spawns some Battery Acolytes, these drop a number of orbs of light, use them to charge your super and fuck up the bastard.
Blind Well Heroic
Beat Tier III and head back to the center with an Unstable Charge of Light, pop it in and face one these big baddies:
Sikariis and Varkuuriis, Plagues of the Well
These guys spawn with invulnerability shields, and the only thing that can break the shields are the screebs explosions that they spawn at their feet. So basically just blow up those screebs as soon as they spawn, then go nuts.
Cragur, Plague of the Well
For this boss, he'll spawn with an invulnerable shield. The only way to hurt this shield is to find the yellow health hive enemies that spawn on one of the plates around the room. Kill those yellow healths, and some purple balls spawn. These balls function exactly as they do in The Corrupted strike: you grab one, pass it to someone who's looking at you, and they throw it at the boss. Once the shield is broken damage the shit out of him, you get 3 chances to break his shield per run.
Inomia, Plague of the Well
This boss functions almost exactly like the boss in the campaign. Occasionally he will go immune, just go and find the taken blight thing that's protecting him, kill it, and continue beating him up.