I don't have numbers or references here, but my intuition from a lifetime of space nerding is that, if the moon were struck with enough retrograde force to materially lower its orbit, in the best case that impact would liberate debris of mass and vector such that our first concern would be planetwide disaster response. (In the worst case, it might shatter the moon entirely and distribute a significant fraction of its total mass across Earth's surface; in such a scenario, the survival of eukaryotic life anywhere on Earth qualifies as a million-to-one-shot win.)
But the odds of such a collision I suspect are significantly lower even than of a similarly sized impactor striking Earth. Granted that the moon helps a great deal in sweeping space near Earth of objects that might otherwise enter our atmosphere, but something with the kind of kinetic energy it would take to produce these outcomes isn't going to spend long enough within the gravitational influence of the Earth-Moon system to be significantly affected before impact. It'd be like trying to swerve an ICBM warhead by blowing really hard on it, and I choose that simile with care - if this ever does happen, I think it might constitute a clear indication that someone somewhere in the universe really hates us!
But the odds of such a collision I suspect are significantly lower even than of a similarly sized impactor striking Earth. Granted that the moon helps a great deal in sweeping space near Earth of objects that might otherwise enter our atmosphere, but something with the kind of kinetic energy it would take to produce these outcomes isn't going to spend long enough within the gravitational influence of the Earth-Moon system to be significantly affected before impact. It'd be like trying to swerve an ICBM warhead by blowing really hard on it, and I choose that simile with care - if this ever does happen, I think it might constitute a clear indication that someone somewhere in the universe really hates us!