Theory time
This theory is in consideration of Law's background with Doffy and Drake's previous affiliation with the marines
this theory assumes that Drake knows some details on the corruption of the marines and specifically the WG - perhaps he found out what happened during the 100 year void somehow - it certainly would explain why he left the marines and became a pirate or perhaps as you see later Dragon tells him. I'll bet on it that Drake got wind of the corruption while in the marines and went to Dragon/revolutionary army for more answers before formulating his opinion and leaving.
Law also knows quite a bit about the will of D. Given his previous association with Doffy, he clearly knows more about the world than he lets on - "what is it you marines are planning" to smoker is a line that comes to mind and also when he mentions that the marines have been stockpiling their forces as if in preparation for something. Law could be part of the revolutionary army somehow. In my opinion though, Law isn't directly associated with the revolutionary army and instead has some sort of a grudge against Doffy and Kaidou which could explain his cooperation as explained later on…
Dragon basically gives Drake the task of infiltrating Kaidou's crew - this seems plausible because imo Kuma is allowing vegapunk to make pacifista's based on himself so that Dragon can make use of them later on (another theory of mine).
Law may have an encounter with Drake and get to be part of the grand plan. Dragon tells Law everything he told Drake about WG/100y void.
-Dragon tells Law to save Luffy (hard to understand why else he would do something so reckless)
The grand plan begins - infiltrate and take down one of the emperors (Kaidou) to destabilize the New World - other pirates/yonkou and even marines will seek to gain dominion over what used to be Kaidou's islands. This gives the revolutionaries some sort of an advantage - with all their opposition focused on the new world, Dragon can take control, err revolutionize WG controlled islands and tell them the truths about WG (ex. telling King Cobra/Princess Vivi about WG's corruption and the 100 year void would definitely sway their opinions towards the revolutionary army's general interests).
Law is given the task of joining the Schichibukai to keep a closer tab on the WG and their plans (the schichibukai seem to be informed of certain future events such as Doflamingo knowing about marineford - and even earlier on, mihawk, kuma and doffy knowing that crocodile has been taken down and that blackbeard has his interests in joining before anyone else/general public knew).
Law would have to make his way to Punk Hazard - get in good with Caesar and await the signal to take down Caesar and screw up the SAD supply causing possible distress between Doffy and Kaidou and even the rest of the brokers (ex. shit will hit the fan if you tell 'Jack' about what happened - maybe Jack is Kaidou's real name).
Law making an alliance with Luffy is simply a huge bonus to the plan (or a detriment knowing Luffy - could possibily cause Dragon to antagonize him if things go wrong)
Law's end of the plan is a success so far
future events:
-Kaidou gets pissed
-while aggravated or even occupied with fighting doflamingo (could happen) Drake Luffy and Law take him and his crew down
-Marines/WG/yonkou/other pirates capitalize on the opportunity of expanding their territory in the NW (a la whitebeards death)
-Dragon and revolutionary army spread their influence everywhere else.
Reading this over it seems really dumb though - think of it as a shitty alternative OP timeline haha