From my understanding, in general, conscripted or volunteer, Lincoln found it more politically expedient to allow new regiments to be created, thus creating new colonels in command (thus owing political favors in return) than to shuttle replacements into veteran regiments. Thus, by late 1864, many of the hardest fighting Union regiments were reduced to skeleton organizations and disbanded. The Confederates on the other hand generally didn't have the luxury of making that choice due to manpower pool limitations. All replacements, volunteers or conscripts, were absorbed into veteran units. I think it is generally conceded that the Confederate system was better at incorporating new men and readying them for combat as opposed to pressing newly raised units "in toto" into action.
I respectfully disagree with this, as I feel it is a myth that doesn't hold up to analysis. Allow me to explain, with the caveat that I am only knowledgable about New York regiments.
The state of New York successfully raised ~185 infantry regiments during the war. Of that number, 177 were raised in 1861 or 1862, with another being raised in 1863. Throughout the war, efforts were made to recruit to veteran units. However, vast recruiting drives - like those seen in 1861 and 1862 with bands and speeches and bonfires (and big cash incentives) - were far more successful than day-to-day efforts.
In my view, there is truth to political expediency idea of putting favorable people in places of high rank. But there was also the appeal of enlisting with neighbors and joining the service as a sergeant or corporal or 1st Lieutenant, instead of joining as a private and having to work up the ladder. (Just my opinion here.)
While New York did disband numerous 2 year regiments in 1863, there were plenty of men in each unit that had signed up for 3 years after the initital surge of enlistment, and others who re-enlisted, and these men were shuttled to other New York that needed bolstering in numbers. This was common practice throughout the war. (For example, in 1863, the 76th New York absorbed about 40 men from the 24th New York, plus men from the 30th New York. When the 76th's 3 year enlistment was up in late 1864, the regiment's re-enlistees and conscripts were transferred to the 147th New York.)
In 1863, enthusiasm for enlistment slackened to the point that a draft was necessary. Thousands of New York draftees were shuttled to New York regiments in the field, bolstering numbers that death, disablement, illness, and desertion had sapped. The 147th New York, for example, received more than 500 conscripts and put more men into the field at The Wilderness than they had at Gettysburg. Furthermore, many regiments recruited over the winter and further bolstered their ranks with enlistees (though not nearly the number as conscription netted).
It is true that many regiments were skeletons of themselves in late 1864. At this point, enlistment drummed up again and new regiments were formed in New York. The incentives were only a year of service and huge cash bonsuses. But this was not all that effective (and did put green troops into the field). I need to look into this more, but I believe there was a draft in New York in early 1865 - but the end of the war prevented men from actually being shipped to the field.
If you look at the total numbers of men who served in New York infantry regiments, the numbers far exceed the initital 1,000 man limits of initial enlistments. This is especially the case of units that enlisted later in 1861 and beyond. The 147th, for example, had more than 1,700 men rotate through the regiment between 1862 and 1865.