MPs don't do anything. MPs are like uniformed patrol officers. They can hold the soldier until his command shows up to claim him or his command puts him into pre-trial confinement.
You are using terms that are not in the Military justice system. There is Non-Judical punishment(NJP), aka Art 15 that can be given by the soldier's Company/Battalion/Brigade/Division Commander.
For Judical Punishment, aka courts-martial, the Company Commander "prefers" charges for courts-martial. The Battalion Commander can endorse or decide on NJP The Brigade Commander, who typically will hold special Courts-Martial Convening Authority, can endorse the charges and send it the Division Commander, or he can keep it, refer the charges to either a Summary or Special Courts-Martial. If the Brigade Commander endorses the charges, as I said, the charges are then sent to the Division Commander who will order an Art 32 hearing, the closest civilian comparable is a probable cause hearing. The Art 32 officer will hold a hearing and then recommends to the Division Commander whether a courts-martial is warranted or not and if yes, the type of Courts-Martial The Division Commander does not have to follow the recommendations of the Art 32 Officer. He can "refer" charges to a Special Courts-Martial with Bad-Conduct Discharge Authority, a Special Courts-Martial, or a General Courts-Martial. He can also send it back to the Brigade Commander for him to handle, or he can give NJP.
If charges are referred for courts-martial, the soldier can request a Chapter 10 administrative separation which comes with an automatic Other Than Honorable (OTH) Discharge.
At no time is the soldier arrested or "charged". The soldier will remain in his unit doing his job throughout the entire proceedings. The Company Commander obviously will revoke any and all automatic pass privileges. A growing trend was to make sure the unit wasn't mistakenly "punishing" the soldier, the Co Cdr would put the soldier into pre-trial confinement, which the soldier can appeal to a military judge. This was typically only done if there is a substantial risk of flight by the soldier or that the soldier is a continued danger to himself or others.