I'm not aware of any systematic investigations. One thing to keep in mind is there's a big distinction between say going to Mars and going to planets in another solar system. Going to a whole other system reduces more existential risks but requires a lot more investment.
It might make more sense to think of this in terms of specific risks and whether they would be avoided simply by having a backup population. Obvious examples in that category are asteroids and nanotech disasters. The first however is more easily dealt with by careful observation and moving the asteroids out of the way. The second is much harder to quantify but many people who actually study nanotech consider it to be unlikely.
Colonies do however also protect against the unknown unknowns. But for obvious reasons it is very difficult to estimate how much they do so.
Can anyone recommend some resources, or provide actual examples, of any relevant evolutionary-biology-style equations which could describe what factors would most likely predominate to allow for the survival of sapience, given such factors as the ease which any given individual can acquire the means of killing large numbers of people, the scale of those means, the willingness o people to use those means, the ability of small groups to travel away from other groups, and so on?
Or, put another way - is there any way I can quantitatively check my intuition that a valuable way to avoid certain existential risks is to flee Earth?