A constant voltage constant current  (CVCS) regulator  doesn’t mean a system with a constant load, since there will be no regulation in such case.  What we call CVCS regulator is a regulator with two modes. The first mode is constant voltage, where the regulator trying to regulate the supply to a variable load at a constant voltage. In this mode off course the current will be variable depending on the load. The current will vary a a range that is limited to a predetermined maximum current level. When the load draw more current than this level then the regulator will switch to constant current mode, where the regulator keep the current at a fixed maximum value. Any attempt of the load to draw more current will cause the regulator to decrease the voltage as a reaction to  keep it constant, so it acts like a current limiter. Here is the schematic diagram of the CVCS regulator.

5 A Constant Voltage Constant Current Regulator circuit schematic