When operating within specs, the output voltage remains fixed to a value at or very close to the specified voltage. Within those constraints the output current can do anything.
If it is a linear regulator, the input and output current will remain equal, except for the current actually required to operate the regulator itself.
If it is a switchmode regulator, the input current will rise as the output current rises, and they will remain in proportion to each other (again after the exclusion of the current required to operate the regulator). However with a switchmode regulator the input and output powers are the same (at 100% efficiency) which means the input current can be higher or lower than the output current; the relationship is determined by the ratio of the input and output voltages.