Disregarding specific examples, I'm thinking there are two variables that determines the difficulty of this reality jumping. One I'll call distance, and one I'll call precision.
Distance is a measure of how far back the split between universes occurred. So a universe where Napoleon won the Napoleonic war is more distant than one where your opponent actually missed you with that gunshot three seconds ago.
Precision is a measure of how many "subevents" that need to line up just right, to arrive at the present that you want. So a universe where your opponent purely misses is less precise than one where he actually hits the guy behind you, which is less precise than a universe where he hits that guy right between the eyes.
More distant universes inherently requires more energy to get an equally precise present. This is because more events need to line up just right, to arrive at the same result. If your opponent shot you yesterday, and you want to end up in the exactly same situation that you are in now, except you don't have a gunshot wound (IE, he didn't hit you yesterday), you are going to need to spend a lot more energy, than if you had negated the gunshot wound three seconds after he shot you.
How difficult you want more distant places/more precise events to be, depends on the sort of campaign/setting you want to have.
With cheap distance but expensive precision, the players can easily jump to the universe where Napoleon won the war, but will have a hard time controlling what exactly is going on in that universe. For short jumps, that might mean jumping so that your opponent didn't shoot you, but instead finding yourself in a universe where you got knocked out the window instead.
With expensive distance but cheap precision, players purely be changing "local" stuff. So no traveling to alternate timelines. Instead the jumpers have a large degree of control over the events of now and a few seconds back. Impossible dodging, incredible luck and other Matrix-style stuff occurs.
With expensive distance and expensive precision, jumping more or less just becomes a reroll. You can change the immediate past, but you have no control over what you get instead.
With cheap distance and cheap precision, you get both the possibility of alternate timelines, and Matrix-style dodging and precision shots. In addition there comes the question of how much more expensive that precision becomes with distance. Can you jump to the exact same situation that you are in now, but in an alternate universe where Napoleon won the war, so that of being a cop, your opponent is now a gendarm and a member of the illuminati? Or the exact same situation in an alternate world where dinosaurs involved into dragons, so in just a sec, a dragon is going to burst in through the roof.
No matter what configuration, I'm thinking of three general moves, one dealing with distant jumps, and two dealing with precise jumps, with some tweaks depending on the exact setup that you want. Generally the moves generally results in less precision, the worse the roll.
Distant Jumping
State what kind of alternate timeline you want to end up in (one changed event in the past)
10+: You get more or less exactly what you wanted (A Franco-dominated world)
7-9: You get what you wanted, but with other major events interfering (Napoleon won, securing Europe, but Japan won the world war in 1928 and subjugated North America)
6-: What you wanted happened, but was completely negated (Napoleon won, but revolutions in the 1880s resulted in a German-dominated European Republic)
Precise Jumping(Defensive)
State one recent hard move that you would want to negate
10+: The hard move is negated, and you get a benefit (Opponent misses you, and hits his ally)
7-9: The hard move is negated (Opponent misses you)
6-: The hard move is negated, but the GM gets to make another hard move (Opponent misses you, but then knocks you out the window instead)
Precise Jumping(Offensive)
Change the result of a recent move you've made.
10+: Treat as if though you rolled 10+ on the original roll.
7-9: Treat as if though you rolled 7-9 on the original roll.
6-: The result of the move you made is negated, and the GM gets to make a hard move.
To keep the changes easy to handle, you might state that Precise Jumping can only be used immediately after the affected move / hard move. That way you won't have to retcon everything that happened between the move and the Precise Jump. A move that happened earlier would then be handled with Distant Jumping, which more easily handles such fudging.
Depending on where you want to put your Distant/Precise levers, you can apply costs to the moves, or remove them outright. Maybe you need some kind of hold to do some or all of the moves. Maybe there are penalties depending on the situation. Maybe the stat that is rolled against changes depending on situation (using some mechanism to charge up your jump device from +0, to +1, +2 and finally +3). Your Reality Collapse of course also serves that.
I also imagine that there is space for a move that could somehow express the insanity that start happening, if two jumpers start altering and counter-altering reality during a fight. That could just be handled by a load of Precise Jumping moves, but another move might be able to streamline that without losing flavor. Haven't quite figured that out yet, though.