You may transport firearms just like any Illinois resident with an FOID card. Since you can not get a FOID as a non resident, the state allows non residents who may legally possess firearms in their home state to transport firearms as long as they are unloaded and in a case.
Ammunition may be in magazines or speed loaders and may be in the same case or container as the firearm, just not inserted in the firearm..
Any case that completely encloses the firearm and has some type of fastener, zipper, Velcro, snaps, clasp etc. does not have to be a hard case. Transport in Illinois does not require any specific destination.