This is the premise for Seveneves, a science fiction book by Neal Stephenson.

If a sufficiently large enough Asteroid impacted the Moon (as 5 million so far have in history of the Moon, the Tycho Crater being half the size of Europe) and the Moon is still in orbit I don't see any feasible scenario. However if it were a Ceres sized Asteroid impacting the moon a few thousand chunks each five times the Everest sized Asteroid that took out the Dinosaurs would undoubtedly hit Earth ripping the Crust asunder, splashing the atmosphere into space and evaporating all of the oceans into the vacuum of space. So are you happy now? ALL life over size of microbes buried deep in rocks GONE.

first immediate effect would be a pretty spectacular view - Really depends on a lot of factors like angle/trajectory/velocity of the asteroid.

But more than likely the first few hours, nothing much - but after a short time tidal waves, extreme flooding, possibly even earthquakes. Hard to say exactly but it won't be good.

Then comes the shrapnel, after a few months there will be a pretty significant increase in impacts around the world, how large they'll be and for how long - I have no idea. But some will absolutely spiral down towards us.

After that our rotation might speed up, a day could last significantly less time - Seasons may go out of whack.

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That's a big asteroid: It's had to dump more than the gravitational binding energy into the moon.

So, the result is 100 km sized chunks of rock flying in all directions. Over time, some will reform a smaller moon, orbiting around a sterile, dead, cratered Earth.

That's because some of the others will hit Earth. Hard. It'll be (briefly) a more intensive period of crater forming since Late Heavy Bombardment. It won't quite boil the oceans and yet it'll wipe out anything more complex than bacteria.

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Hello u/planktivious, your submission "An asteroid flies past Earth and totally takes out the moon at 5AM EST like it was never there. What happens in those first few moments all over the globe? And what eventually happens overall?" has been removed from r/space because:

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Tidal forces all gone as debris moves away from impact. Earth rotation may be affected by loss of moon resulting in catastrophic orbital wobble (poles go to equator and vice versa). Debris rains down on the Earth. Meteor storms. Life in the oceans not finding tides will greatly be impacted like no reproduction; thus, massive die off of species. Magnetic field may be impacted by orbital wobble.