As with any military equipment, rifle technology improved rapidly during the war. A variety of breech loaders entered the service, primarily on the Union side. These rifles could be fired very rapidly and also reloaded rapidly. The rate of fire increased significantly at this point, but it was too much for the military hierarchy. President Lincoln had to order the adoption of the Spencer breech loader, and as soon as the war ended, they were retired. The Army was out of the assault rifle business until the adoption of the .30-40 Krag thirty years later.
This didn't slow down the development of rifles, though. The civilian market, and foreign military markets, encouraged the development of better rifles.
The late 1800s saw the development of improved lever, bolt and pump action rifles. In the various Franco-German wars, the advantage shifted from one side to the other with the adoption of new and improved, mostly bolt action, breech loading rifles. (And other equipment.)
By World War I, the armies of the world were armed with bolt action rifles that were durable, accurate, and long ranged, but they were not assault rifles. They could not achieve a high enough rate of fire to keep defender's heads down. The machine gun, with its high rate of fire, was far too effective a defensive weapon. Bolt rifles could not attain a rate of fire high enough to support an attack, and machine guns were not portable enough to carry with the attacking troops. The war became a defensive stalemate that dragged on until the civilian populations were too exhausted to support it any longer.
Towards the end of the war, after the blood letting of Ypres, the Somme, and other human wave battles, the need for light weight automatic weapons became apparent. Italy had been the first country to adopt a submachine gun, in 1915, but it was not widely used. By 1918, Germany had introduced the Bergman MP18I submachine gun, and the United States developed the Browning Automatic Rifle. Both saw some combat, but arrived too late to significantly affect the course of the war. The Pederson device was a separate action (semiautomatic) that replaced the bolt in a U.S. Springfield bolt rifle, but it arrived too late for service in the war. None of these efforts quite made the grade as assault rifles. The submachine guns were under powered and had inadequate range for open terrain. The BAR was heavy, which slowed it down. The BAR man had to struggle to keep up with his squad and this limited the amount of effective fire he could lay down.
