During a war, an AI increases its chances of building units. That claim is a fact that is based on several people having code-dived. Apparently, Missionaries are included in this category (units), or at least the "Missionary Spam" Civ-wide mission isn't excluded from occurring during wartime.
Rarely do you see an AI interrupt a World Wonder build during peacetime. Once a Wonder gets started, it usually stays in the build queue.
The opposite is often true in wartime. Often you will see an AI switch from building a Wonder partway through completing it... and then it seems to randomly decide when to build said Wonder again... to the point that it doesn't necessarily "remember" that it had started on the Wonder in City X and can easily start to build the Wonder in City Y and then switch to building an Archer (or in this case a Skirm) partway through building the Wonder in City Y. You'll sometimes see that an AI has more Gold for trade after having completed a Wonder when it had been doing a lot of warring for this very reason (the AI earned itself some Failure Gold by accident).
Being at war messes with an AI's ability to start and complete World Wonders.
We have a good 10 to 15 turn window where we won't be able to get Literature in trade.
Let's say that we get Aesthetics for Metal Casting but don't bribe Mansa into a war. Well, he's not powerful enough to start a war on his own, while no other AI is likely to declare war on him (Dr. Evil would probably be the only potential candidate, but Dr. Evil won't plot a new war while he's at war with us).
So, anytime during that time, Mansa can start on The Great Library, meaning that we won't get Literature in trade from him. Sure, we could THEN bribe him into a war for Horseback Riding, but we'd still have to self-tech Literature, as we wouldn't have a tech to trade to him for Literature.
If he's involved in a war, he has a lower chance of building Wonders.
If he's involved in a war and he starts to build a Wonder, he has a much greater chance of abandoning that Wonder mid-build... all that we need is a 1-turn window where he stops building The Great Library after we have learned Horseback Riding and we can get Literature from him in trade.
It's also obvious that an AI REXes best when it has more Military Units and doesn't dump as many Hammers into Wonders. Getting an AI into a war, which code-diving has proved increases an AI's odds of building a unit, therefore does help an AI to REX better.
On turns where Genghis (or Joao) is at war with both of us, there's nothing to say that Genghis (or Joao) will send any less units our way than they would have otherwise done... it may or may not happen.
HOWEVER, on the turns where we take a Cease Fire in between Worker steals, we CAN count on Genghis' (or Joao's) stack to head toward Mansa. It won't sit around camping near our empire. That's something which we can count on.
And, in between Worker steals, we have been Cease Firing, so such a period of time will exist.
What's not clear is, once an AI starts sending its stack toward Mansa, and we redeclare war a couple of turns later, whether said stack will keep going for Mansa or will come back toward us... but minimally, we'll have bought us a couple of turns with the stack walking in the wrong direction... so, we'll have double that amount of time from which we'll be safe from the stack (if it walks away from us for 2 turns, it will need to spend 2 turns walking back toward us if the stack chooses to beeline us at that time).
I have no clue how to tell which units Genghis has. All I know is that if I look at the F5 -> STRATEGIC ADVANTAGES screen, Gandhi, Genghis, Joao, and Shaka all have "Chariots" listed as "Their Units -> Possible," while Mansa does not have Chariots listed but Mansa does have Skirmishers listed under "Their Units -> Known."
After we Cease Fire with Genghis, we can ask him repeatedly until he brags about a particular type of unit, but until that time, I don't know of a good way to tell which units he can build, and I don't have any clue of how to figure out which units he actually has.
An AI will not Declare War unless it can build 2 "non-Archer" types of units. However, it CAN send in stacks of whatever units it has available if declared upon, be they Archers, Chariots, a mix thereof, or even Warriors at lower difficulty levels.
Assuming that the F5 -> STRATEGIC ADVANTAGES screen gives us a complete-enough picture, Genghis doesn't have Copper anywhere else. If that's true, the only thing keeping him away from getting Spearmen is us continually stealing Workers that attempt to hook up his Copper Resource.
So, attacking him is not just about getting free Workers, but it's protecting our future investment in Horse Archers and War Elephants. It also means Genghis not building a stack of Axemen, Chariots, and Spearmen to come after us.