The part that hasn't been explained is the fact that they are accelerating away from one point. There is nothing beyond them increasing there speed. The whole slingshot effect is made by an object _falling_ towards a gravitational pull. Much the same as something orbiting a planet , however they accelerate through the gravitational pull.
Secondly the field of gravity has to have some sort of measurable speed. For instance if another planet was instanteously set in our solar system there would be a time where it's gravitational pull did not effect the earth, however minute it may be, there would still be a time from the instant creation and the development of the field. (Say god did it.)
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Errrm the relative distance between most galaxies is such that the effect of gravity from one galaxy on the next closest is realtively negligible and they are moving in the best vacuum that exists...space is very very big...The force that propelled them was immense beyond comprehension...In fact the radio waves sent out by the explosion are still audible (Penzius and Wilson) won the Nobel for identifying this radiation...
Secondly gravity is a force that is constantly exerted by every object in the universe on every other object in the universe it increases and decreases based on the relative distances between any 2 objects...so if matter were to appear instantly (which it can't within current physical theory) at x distance from another bit of matter it would instantly exert a predictable amount of force as far as we can determine...
Its a force not a particle or a wave form...

m1 and m2 are the masses of the 2 objects , d is the distance between them and Fgrav is the force of gravity in Newtons
Here is a nice concise view of the Big Bang Theory. It also contains a discussion of the previously popular Steady State Theory.
Here is a nice overview of Hubbles Law of The Relative Motion of the Galaxies. See also Red Shift and The Hubble Constant.
Basically as massive as the Galaxies are...they are so far away from each other for the most part that their relative gravitational forces on each other are have negligible impact on their accelerations...
Galaxies within galactic clusters do have gravitational effect on each other so you won't if looking at a galaxy in a "local" cluster notice any relative acceleration (eg M31 Shows no Red Shift) we are therefore accelerating with M31 (common direction and acceleration).
I hope that makes a bit more sense...
...
Secondly gravity is a force that is constantly exerted by every object in the universe on every other object in the universe it increases and decreases based on the relative distances between any 2 objects...so if matter were to appear instantly (which it can't within current physical theory) at x distance from another bit of matter it would instantly exert a predictable amount of force as far as we can determine...
Its a force not a particle or a wave form...

m1 and m2 are the masses of the 2 objects , d is the distance between them and Fgrav is the force of gravity in Newtons
Here is a nice concise view of the Big Bang Theory. It also contains a discussion of the previously popular Steady State Theory.
Here is a nice overview of Hubbles Law of The Relative Motion of the Galaxies. See also Red Shift and The Hubble Constant.
Basically as massive as the Galaxies are...they are so far away from each other for the most part that their relative gravitational forces on each other are have negligible impact on their accelerations...
Galaxies within galactic clusters do have gravitational effect on each other so you won't if looking at a galaxy in a "local" cluster notice any relative acceleration (eg M31 Shows no Red Shift) we are therefore accelerating with M31 (common direction and acceleration).
I hope that makes a bit more sense...
