In military terminology, this would usually be called replacements. Reinforcements would mean an additional unit added. So an infantry division would be reinforced with a task battalion, but the group of riflemen sent to take the place of casualties would be replacements.

Some replacement would happen organically due to the activities of unit medical personnel. Exhausted soldiers given bed rest and hot rations return to duty quickly. Minor injuries can be treated and rhe troops return, often within 72 hours.

In some cases stragglers and men who got separated from their units would be scooped up and temporarily incorporated into units that needed them. The 6th Armored Infantry Regiment basically fell apart in the assault on Mount Porcia. When the 48th Engineers attacked again, they collected several hundred stragglers and lightly wounded men in the process.

But dead and seriously wounded troops had to be replaced. In the US Army, large numbers of men were given infantry training and shipped to Europe or the Pacific to fill in depleted units. Ideally, the unit would be pulled off the line to incorporate these men properly and do some training, and this was often the case when preparing for major operations such as amphibious landings.

Frequently, however, this system couldn't keep up with the demand for riflemen. On several occasions, US generals stripped troops out of support units in order to feed the front. These troops were not properly trained for rheir new roles and as such took higher casualties than proper infantrymen would have.

A good media depiction of the effect this system had on unit cohesion is in the movie Big Red One, where the veteran soldiers inform their replacements that no one wanted to learn their names until they proved they would survive a fight.

About the Soviet system I am less familiar, other than the fact that rifle divisions of 2,500 men were not uncommon. That suggests their system was less efficient.

The Germans had a good replacement system until it was overwhelmed by high casualties and poor transportation networks in Russia, but further than that I can't say.

Specifically for the US, replacements were trained at one of nine Infantry Replacement Training Centers known as IRTCs or simply RTCs. RTCs were established at Camp Blanding FL, Camp Croft SC, Camp Fannin TX, Camp Hood TX, Fort McClellan AL, Camp Roberts CA, Camp Robinson AK, Camp Wheeler GA, and Camp Wolters TX. Training eventually settled on a 17-week course, although the curriculum would continue to change throughout the war based on feedback from Army Ground Forces (AGF) and theater commanders.

Replacement training had expanded from 13 weeks to the full 17-week course after North Africa had proven existing training was insufficient. The expansion included several combat courses and an additional two weeks of unit training. Combat courses were to infuse realism into training and included an infiltration course, close combat course, village fighting course and later subjecting them to being near incoming artillery fire. The unit training inexplicably centered on the company level. AGF resisted calls to include training at the regimental level, and training regiments were not T/O units, consisting solely of rifle companies with no heavy weapons. Feedback from Europe was that company level unit training was really too high-level, and the time would be better spent on squad and platoon level training.

There was always a tug of war between creating new divisions and providing replacements for existing units. AGF Commander Leslie McNair and Chief of Staff George C. Marshall were firm advocates of the replacement system rather than creating new units once they reached around 100 divisions. The argument for new divisions, which was championed by Secretary of War Henry Stimson, intensified in late 1944 during the Battle of the Bulge. Marshall managed to convince Stimson that the replacement system was sufficient because it took so long to train a new division and the Selective Service system was struggling to meet the manpower requirements of the replacement needs. Division training had been reduced from 44 to 35 weeks, but there was as much as five months of lead time to select and train key leadership positions, junior officers, and NCOs where they were ready to receive recruits from the reception centers. The replacement system was never perfect but a division with 100% of its manpower and equipment and 80% combat veterans was more effective than a division with 80% of its manpower and equipment and 100% veterans.

I have less information on how replacements arrived in their units once they had arrived in theater. The ideal situation would be to introduce replacements while a unit was in reserve, but that wasn't always possible. During the crisis of the Battle of the Bulge, there were over 200,000 US replacements available in Britain and France. The problem was the Allies had dropped the ball on opening the port of Antwerp and supplies were coming from ports in northwest France. This meant that when the battle began, replacements (as well as reserve divisions) were hundreds of miles from the front where they wouldn't further stress supply lines. Also, while you may be able to introduce replacements while they are entrenched on the front, it's almost impossible to do while a unit is actively engaged in combat.

The most extreme example of the replacement system working is the 105th Infantry Regiment, a National Guard unit from New York. The 1st and 2nd battalion of the 105th suffered 80% casualties over a 24-hour period on Saipan. Some of those casualties would later return after recovering from wounds several other members would transfer out to other units. Nine months later they had received replacements, and while still slightly understrength spent two weeks in heavy combat on the Shuri line in Okinawa. Experienced NCOs and good officers with a majority of green troops could create a very competent unit.

Sources:

Palmer, R. R., The Procurement and Training of Ground Combat Troops

Pogue, F. C., George C. Marshall: Organizer of Victory 1943-1945