As best I can tell, the only organizations actually transferred to Confederate service on Mar 1, 1862, were the Orleans Guards Battalion, the Crescent Regiment and the 5th Company, Washington Artillery. Under advice of Secretary of War Benjamin, these organizations were accepted under the Local Defense law of Aug 21, 1861, for a term of six months. At this time (Mar 1, 1862) the War Department was attempting to assemble an army at Corinth MS for General Beauregard. See OR, Series I, Vol 6, p 837. All these commands immediately went north to Beauregard.

For other militia organizations the transfer to General Lovell's command had little impact. The militia he received from Governor Moore didn't "become" Confederate soldiers as a fait accompli. They continued to report to militia brigadiers like General Butler, who are not in the list of Confederate general officers, with no need to conform to Confederate regulations as soldiers had to do. Lovell continued to ask the War Department for Confederate units to defend New Orleans and complained about the quality of the militia in the city. On the other hand, these militiamen were entitled to pay at the end of their terms that came from the Confederate Treasury Department.

Something else to bear in mind --
After Governor Moore called these men into service on Mar 1, 1862, the Confederate Congress passed the law of Apr 16, 1862. As you know this law required all men within the age range 18 to 35 to be enrolled in Confederate service for three years or the war. That applied to all militia formations in service or not. Since most members fell into this age category, the law effectively put an end to their militia careers. For example, the members of the old Sumter Regiment should have served out their ninety-day term. Since the law required them to be in Confederate service by May 15, 1862, or face compulsory conscription, most complied.

For this reason we find the organization date of the 30th Louisiana Regiment to be May 15, 1862, the Sumter Regiment disbanding prior to end of its nominal term.
The same treatment was given to members of the three organizations named above in Bragg's Army of the Mississippi. Most members of the Crescent Regiment and Orleans Guard Battalion were assigned to the 18th Louisiana Regiment, supernumary officers being sent home to face conscription.

If you'll pardon the expression, the Confederate army was no place for militiamen.

Scene from 'No Country for Old Men'