In a DC motor, how is it that field flux can stay constant despite a changing load?
In the shunt configuration, the armature and field voltages are the voltage of the supply. If the supply voltage is held constant, then it does not change with varying load. If the supply voltage is not held constant, it will vary due only to the voltage drop across the internal resistance of the supply. If that resistance is small, as it should be, the voltage variation with load will be small. The voltage drop across the armature resistance will cause the back emf to change and the speed will change accordingly, but the field voltage will be unaffected.

In the series configuration, the armature current also flows in the field, so the field flux changes as the load varies and can not be kept constant.

If the field is provided by a permanent magnet or separate power supply, the armature voltage can be changed without affecting the field flux. The field supply voltage can be changed to change the field flux without affecting the armature voltage.