Is your deployment unit assignment done prior to deploying or once you are on site?

As others have mentioned, usually it's done prior to your movement. However, in your case I expect that it is the same as what was done in DS/DS - there will be a mixture.

For the "I need a left-handed basket repairer", there will be a specific assignment to a specific unit. However, there will be a lot of assignments where you are deployed to a region and once you arrive, the personnel folks will determine where you go.

As MSG (Join to see) pointed out, even if you have a directed assignment, it can change once you're on the ground because of shifting priorities.

I believe it depends on the deployment. Mine were always done ahead of time.
MSG (Join to see)

During the Viet Nam war, all but the first units to go, were actually individual PCS moves. To answer your question, you could have been sent wherever the replacement battalion had a need that day. I know this personally. After filling sandbags all day at 90 replacement, I came down with heat exhaustion and missed a roll call. Had I been there, I would have gone to the the 23rd ARTY Grp. Being it was thenext day, I went to the 25th INF Div.

Normally, assignment is done prior. However, once on ground that could change depending on the needs of units. Especially in that Era. So, really, the only answer is that it all depends

Since Vietnam, the Army went to unit deployments vice individual filler systems. A Combatant Commander requests a capability and the joint forces requirement generation process tasks it to a service and a unit. That unit receives orders to deploy. So individual soldiers deploy with their assigned unit. There maybe case but case circumstances where personnel are assigned from other units to fill vacancies of a deploying unit. Our late joining replacements deployed through our home station rear detachments, received predeployment training and equipment, then joined us down range. The Division Rear D might shuffle some folks to fill urgent needs, but that person is pinpointed to their unit.

All the transport shenanigans, are done as a unit. Many units deployed as a unit to Vietnam as well. Then they left the guidons and equipment there and rotated people. Now the whole unit comes back as a unit. Sometimes with their equipment, sometimes not.

There are a handful of individual deployers that are individually ordered to deploy by the WIAS system to fill deployed units such as newly formed headquarters etc that were stood up in theater. Usually filling slots in a Joint Manning Document (JMD) unit or in country special task force. These are very few compared to the thousands of guys assigned to MTOE units deploying as units.