Actually, it was muskets -> percussion caps (introduced in 1820s, but one of the prominent wars to feature this was the American civil war) -> various cartridge rifles, such as breech action and needle rifles (some of which were used during the civil war) -> bolt-action.

Very general list, and I'm probably forgetting a few. However to answer your question of "when was bolt-action prominent?" I say during the time of the Civil War, actually. Not used across the world but cartridge rifles, like I mentioned, was used during the Civil War and one of the winning factors for the Prussians during the Austro-Prussian war of 1866 was the "Dreyse Needle Gun", which was a breech-loading bolt action rifle. The Austrians used a percussian cap Lorenz Rifle which was too slow compared to the Prussian rifles.