
Reviewed for patch 7.3d

Favorable matchup
Reviewed for patch 7.3d
Against Draven, Ezreal refuses the long trade. Mystic Shot punishes him from range every time he walks to catch an axe, and Arcane Shift takes Ezreal out of Stand Aside's line. Without takedowns, League of Draven never cashes in its bonus gold.
Draven has to catch Spinning Axe to keep his damage up: the axe bounces off the target and flies up, and he has to go get it. Mystic Shot from max range makes him pay for that step, and every hit lowers your cooldowns.
Stand Aside throws his axes, knocking aside and slowing whoever they hit. Arcane Shift teleports you to a nearby location and fires a homing bolt at the nearest enemy: keep it to step out of that line or out of Draven's Blood Rush engage.
League of Draven grants Adoration for catching axes and killing minions, monsters or towers, and turns it into bonus gold when he kills a champion. Since 7.3, Whirling Death also executes champions whose health is below his Adoration stacks. No takedown means no bonus gold, but Adoration keeps climbing with every axe and minion: don't sit at low health in his range.
The whole laning phase is played at range. Poke with Mystic Shot when Draven goes for an axe and back off when he pops Blood Rush: with that attack speed, staying isn't worth it.
If you're playing Draven, catch your axes without stepping into Mystic Shot's line and look for the trade once Ezreal has used Arcane Shift. With your support's crowd control, Stand Aside and two axes are enough to push him back. Don't chase past his tower for one axe.

Favorable matchup
Reviewed for patch 7.3d
Draven tells you where he's going to be: under his next axe. Varus turns that into a target with Hail of Arrows, which slows and cuts healing on the ground, and with Chain of Corruption, which immobilizes the first champion it touches.
Spinning Axe bounces off the target and flies high into the air; if Draven doesn't catch it, he loses the next one and the Blood Rush reset. You know where he's going to move before he does, and that's where your abilities go.
Hail of Arrows deals damage and desecrates the ground: anyone standing on it is slowed and gets less healing and regeneration. Cast on the axe's landing spot, it makes Draven choose between catching it slowed and hurt or letting it drop.
Chain of Corruption immobilizes the first champion hit and spreads to other nearby champions. If Draven comes in with Blood Rush looking for the kill, the chain locks him in place in front of your support.
As soon as you have Hail of Arrows, punish every axe catch. With Chain of Corruption up you have an answer to his engage; without it, don't take long fights against Draven and his support.
Playing Draven into Varus, if Hail of Arrows lands on an axe, let it go: losing one costs less than catching it slowed and without healing. Juggle a single axe while Varus has his abilities up and only go in with Blood Rush after he has used Chain of Corruption.

Execution matters
Reviewed for patch 7.3d
Jinx has an answer to Draven's mechanic: Flame Chompers! root the champion who steps on them, and the spot where an axe lands is a good place to drop them. The lane, though, is decided in the first levels: if Draven and his support catch her there, she loses it.
Flame Chompers! bite and root enemy champions who walk over them, and explode on their own after 5 seconds. Thrown on an axe's landing spot, they make Draven choose: step on them or lose Spinning Axe and the Blood Rush reset.
With Fishbones, the rocket launcher, your attacks gain range and deal area damage at the cost of mana. It lets you hit from farther away while Draven catches axes; Pow-Pow, the minigun, gains attack speed and works better when he isn't in lane.
Zap! damages, slows and reveals the first enemy hit. Against Blood Rush's speed, that slow is what buys you time to step back behind the Chompers.
The condition is levels 1 to 3. That's when Draven and his support look for the kill and Jinx has no dash: play behind the wave and give up minions before health. With Flame Chompers! and Fishbones in use, you have more tools for the lane.
If you're Draven, the risk against Jinx is every catch: if she drops Flame Chompers! on the landing spot, let that axe go and play with one for a moment, because getting rooted in front of Fishbones costs more. Go in with Blood Rush after she has spent Zap!, not before.