I asume by connecting in series you mean connecting the GND for sensor 1 to the output of sensor 2, the GND for sensor 2 to the output of sensor 3
If you measure the voltage on the output when this NPN NC proxy is activated, you will measure the supply voltage. This is because most of these proxy's have an internal pullup resistance. You power these proxy's using one PSU. Just make a drawing for the 3 proxy's and their pullup resisance the way you want to connect them in series. Maybe that sheds some light.
The current of sensor 1 and 2 would flow through the output of sensor 3 and the pullup current of all 3 sensors would flow through the output of sensor 3. That whould exceed the max current of sensor 3.
Proxy's aren't the same as ordinary switches and you can't connect them in series but you can connect them parallel.
If you want to use 1 input for all proxy's, you need NO proxy's connected parallel (al the outputs connected together to the input of the board).
Beware that the input on your board should handle the 12 Volt (proxy pullup currents) of all 3 proxy's simultaneously! If the board is designed for 5 Volts, you need a resistor (optocoupler) or diode (digital input).