In most cases, no, although it did vary by unit and circumstance.

Newly-activated or federalized infantry divisions trained for 44 weeks up until November 1, 1942, when that period was reduced to 35 weeks. This was extended to 38 weeks by the middle of 1943, with the training period followed by several more months of maneuvers and various divisional elements being detached for specialized training. During this period, transfers, promotions, and discharges were made, and men were stripped from divisions (the 76th and 78th in early 1943, and 17 other divisions during the middle of 1944) to be sent overseas as replacements.

Examining some of the first morning reports of the 134th Infantry Regiment, 35th Infantry Division, (a National Guard infantry regiment made up at federalization of men from eastern Nebraska) which show casualties, we can see that most men of the regiment in fact were not from Nebraska; men with an Army Serial Number beginning with "20-" were federalized National Guard troops, while men with a serial number beginning with "3-" were draftees. Interestingly, this particular morning report has many men with Army Serial Numbers beginning with "37-", indicating that they originated from the seventh Service Command which presided over Colorado, Iowa, Kansas, Minnesota, Missouri, Nebraska, North Dakota, South Dakota, and Wyoming; how many of these men were actually drafted from Nebraska and happened to be assigned to the 35th Division at some point before its deployment is unknown without further investigation.

At the beginning of the war, the already-active divisions of the Regular Army (1st-9th and 25th Infantry, 1st-5th Armored, 1st and 2nd Cavalry) were made up of career soldiers and draftees. The divisions made up of National Guard troops (26th-38th, 39th-41st, and 43rd-45th Infantry) were often nowhere near full-sized (~14,200 men) when their organic units were fully brought together, and needed to be filled out by activating the remaining units that "made" an infantry division and acquiring large numbers of draftees and volunteers. The first divisions of the Army of the United States (10th Mountain, 23rd, 25th, 42nd, 63rd, 65th, 66th, 69th-71st, 75th, 92nd, 93rd, and 106th Infantry, 11th, 13th, 17th, and 101st Airborne, 6th-14th, 16th, and 20th Armored) were formed completely from scratch (there were exceptions) utilizing transferred Regular Army officers and large numbers of draftees and volunteers. Divisions of the Organized Reserve (76th-81st, 83rd-91st, 94th-100th, and 102nd-104th Infantry, 82nd Airborne) were maintained with all their officers during the interwar period, ("dumping grounds" for newly-graduated ROTC cadets, if you will) but only one-third of their enlisted men; again, draftees and volunteers from all over the United States were used to fill the gaps, demonstrated by the barracks scene in Hacksaw Ridge.

Men trained specifically as overseas replacements would depart in groups (jumbled together at ports of embarkation from various camps) based upon their assigned specialty, having no unit affiliation whatsoever until picked up at a replacement depot. A proposal to improve morale was made in early 1945 by earmarking groups of men for assignment to specific units while still in training in the United States (this part of the proposal was rejected) and have men be shipped in small groups; three groups of four men would form a "squad," four squads would form a "platoon," and four platoons would form a "company" (this part of the proposal was accepted, although it was implemented so late it is hard to measure its effectiveness). Companies and platoons would be broken up as necessary, but the basic group of four men would always stay together, from training to final destination.