The battery voltage and current relation
The output current (and for that matter, the voltage if you consider a battery with internal resistance) are determined by the combination of the source and the load, not by one or the other alone. If you use load line analysis, then you can find the voltage and current from the intersection of the battery's IV characteristic and the load line (the reversed IV characteristic of the load). 

If the load is resistive, like a lamp, then discharging the battery will reduce the output current as well as the voltage.

But if the load is a constant current load (like a battery discharge testing machine) then the load current, naturally, stays constant.

If the load is a constant power load, like a switching power converter itself loaded with a resistive load, then the output current could actually increase as the battery is discharged.