How would all of this vary between a world cut off during the Long Night before being rediscovered and uplifted (for want of a better term) in the Third Imperium, versus a world settled by Imperial citizens funded by wealthy Nobles who in effect are the executive of whatever form of government has been established?
On that same vein, corporate government, I'm the CEO of a megacorporation which colonizes a planet, spends bajillions in infrastructure, offers huge incentives for employees to travel and live there all so I can harvest/refine/manufacture whatsit rocks... now I'm going to let them leave? Hells no.
I'm just going to negotiate with the Imperium a high tax rate or a large one-time payment, maybe some freebies or rights ceded to the Imperium like that asteroid field over there, for exemption from the draft.
Now, can I stop second-generation folks from joining up? Depends on how big my payment was... and whether the Imperium is at war.
===
In my opinion, the real issue is the question of a draft and its place in history. Rebellion and earlier FFW historical "documents" were written in the 70's-90's where a draft was a very real thing. Nowadays, it may exist, but no modern (US) commander ever wants to implement a draft again. And they certainly don't want to command a draft military or take them into action!
In the future, at the Imperium level, you likely want motivated, volunteer military and would not be interested in a draft system or unwilling recruits. Granted, planetary armies and surface conflicts may work differently (after all, some still have quadruped cavalry) but I highly doubt the Imperium would implement a draft.
Canon notwithstanding, an Imperium or Shattered Imperium faction-wide draft doesn't seem very plausible. On a sheer implementation scale, it would be bigger than a census bureau and need more muscle.
You could make an argument that, when called on, a sector must put forth "X units of heavy infantry, X divisions of grav cavalry etc." That's a muster and would be at the sector duke's implementation. Chances are they would fire up the recruitment drive and stations and, if that failed, they would start hiring mercenaries or negotiating a scutage to the Imperium.