The delay function operates as you would think; that is, it keeps the relay closed for some adjustable time after the train clears the sensor. I looked at your Azatrax description and it would be what they call the "Continuous Mode".
I did not explicitly mention it because if ambient light is triggering your sensor, the delay setting would appear to operate backwards too. The relay would click some variable delay AFTER you blocked the sensor window. So the delay would appear to be a delay-before-trigger rather than the expected delay-after-trigger.
Of course what's nice about the MTH and Lionel products is they are packaged whilst the Azatrax modules appear to be "loose" LEDs and photo-sensors that you must mount and disguise. If you are using an Arduino it could easily handle the delay/drop-out timing. And if you don't mind the "loose" transmitter and receiver components, why not use a $1 (and up) eBay IR transmit-receive module rather than the $20 Azatrax module? If using an Arduino, you have 5V DC available which is what these eBay modules use. And if you are using the transmissive mode (breaking a beam across the track), you can just buy loose IR LED and IR phototransistors for, say, 25 cents for a pair. I've written about these techniques in several OGR threads which I can dig up if this is of interest.
