Short answer: probably not usefully.
You're probably going to end up with a molten space-puddle in the process, which in turn is going to take billions of years to cool, the time required to throw enough kinetic energy at it to get it going in the first place, and then doing it all again to slow it down not withstanding. I can't make an estimate of how long that would take because I know nothing about orbital mechanics, but at blind guess, millions of years minimally when you consider how vast space is, how many objects are going to be involved, and the fact that none of them are under power.
But even then there's the problem of having enough rocks around to do this, and to do it with any synchronicity. Remember our own moon was another planet hitting Earth, and it barely effected our orbit at all, mostly just the direction of spin. 921
You need a lot of mass hitting your planet (and if that mass isn't removed somehow, each hit makes the net process harder and harder—the end state of your planet is going to have a helluva lot more gravity than it started with). You probably want many, many, millions (probably billions, maybe trillions; I don't have a real sense of the maths here, and the size of the rocks is really going to matter) of kinetic impacts all timed to gradually achieve your goal. But let's say millions; millions of destination objects are going to add up to many, many more billions of smaller objects involved in these chains of gradually scaling impacts; and not just untold billions of them, but billions of the right size and right trajectories to make this all possible, which in turn means they'd need to be selected from trillions and trillions of objects that weren't useful (or you'd have to explain why everything in this region of space was so auspiciously arranged, anyway). This is turn is going to raise the question of how a technological society emerged in a region of space this ludicrously dangerous in the fist place.
A lot of computing power is one thing, but the society that could even power the computers that could do this math have much, much, much better ways of achieving the same end.
It's also worth noting, though, that this basically is how everything in the universe is currently arranged—gravity and kinetic energy are mostly how everything's moving anywhere as it is.
Sorry if that's not a very fun answer! 921