Max current in Analog multiplexers with latch
supply current specs, towards the top, with the maximums, 


50mA max from VCC
50mA max into GND
+-20mA max from VEE
+-25mA max through the common pin


This is an analog switch, It connects 1 analog input to the common depending on the 3 digital inputs, it updates what input is connected to the common when you toggle the latch pin, 

To take your question at face value, you can source up to 25mA through the common pin, if either an input or the common exceeds the supply voltages, it will flow through clamping diodes on to the supply rails, in these cases you need to make sure that VCC and VEE current limits are not exceeded

Its sounding like you want a device where you tell it what pins should be, e.g. high or low, and you would like it to drive them and supply current, if this is the case a load switch / serial in parallel out shift register sounds like what you are after.