Meteor showers happen when Earth passes through the debris left by a comet in its orbit. Meteors are generally tiny, like snow flings or pebbles and don't make it to the ground as meteorites. But periodic comets sublimate and disintegrate. Don't they shed kilometer sized fragments too? Maybe it is not only the comet which is the threat, but its entire orbit is a minefield too? Are bolides and observed meteorite falls more frequent during meteor showers?
The American Meteor Society FAQ page notes that about 95% of observed meteors are probably of cometary origin. But they're apparently so fragile that they're only about 1/3 of observed fireballs, and 0% of known meteorites.
Also, since the Earth passes through streams of cometary debris several times a year, and yet there's been only one plausible mass extinction due to cometary or asteroid impact in the last few hundred million years (the one that did in the dinosaurs), it appears to be extremely unlikely for cometary debris to be a meaningful danger.
This recent paper by Napier et al. indeed concludes that centaur comets break up into many pieces large enough to cause mass extinction events on Earth. Since objects orbit in the same way regardless of mass, I suppose that dangerously big comet debris are more common in meteor streams, and that major impacts are more common during meteor showers.
Small meteoroidal fragments do have their trajectories changed by Solar heating and the Solar wind, but since Earth crossing centaur fragments seem to be cleared out within only thousands of years, there should be no major difference between the large and tiny objects' orbits.
It also seems as if mass extinctions might be caused by meteor showers depositing dust particles in the atmosphere. Suggested by Klekociuk et al.