It's not unusual for the assistant to conduct the march or one of the "fun, flashy" pieces at a spring or Christmas concert with a few caveats.

The culture of the program is solid.
The assistant has proven themselves OVER TIME to be an integral part of the education of the ensemble.
The serious, big works are still under the direction of the head director.
This also only happens when the HEAD DIRECTOR is ready, not on the assistant's terms.

It's also not unusual for the head director to hand the marching program over to the assistant, provided that the assistant has proven they have the qualifications and skills to run that portion of the program.

However, with what's being described here, new to the program, and being vocal about conducting the group, there probably needs to be a conversation about appropriate lanes for their position.