Larger capacitors hold their peak voltage better till the next pulse than smaller ones do at the same load current, so the charging current changes from a low value long pulse (as a smaller cap charges back up to peak voltage) to a higher value, shorter pulse (as the larger cap charges back less voltage to its peak). In other words, the larger the caps, the smaller part of the total voltage wave contributes to the charging process, so the magnitude of the current has to be higher during those shorter times, to transfer the same average current. This applies to all capacitor input filters, not just those in voltage multipliers.