I have to confess that I am surprised that this may be a revelation for some centers. Granted, it may be a policy issue and not a dispatcher issue, but it seems like common sense:
(Part 1) Verify that the units you have sent to the assignment are actually going.
(Part 2) If they are not responding in a pre-determined amount of time, replace them!
This should not require permission from an IC or some sort of special papal dispensation. It is truly a matter of common sense. The next step, after replacing the missing in action unit, is to notify the responding IC or the other unit (if there is one) that you were unable to verify the other unit and have replaced them with Unit XXX.
You assign two patrol units to a disturbance. The first unit acknowledges the call on their MDC. The second does not. Ten seconds later, you call the unit on the radio and they do not respond. Call one more time. Wait ten seconds. Then assign the next closest car. Immediately after, notify the first unit of the change. This is a safety issue that may impact the actions of the first car. Whether their back-up is coming from one minute or ten minutes away could have a significant impact on whether they go directly into the location, or wait.
You dispatch three engines to a fire along with a Battalion Chief. The second due engine does not acknowledge their run. You must replace them, AND notify both the IC AND the second due engine of the change in assignment. In many areas, the second due engine is responsible for water supply. The third due has a different role on the scene of the fire. Disaster could ensue if the third due engine is not aware that they must now assume the role of water supply. Granted, your procedures may be different, but this is just one situation where replacement and advisement are very important.
The same situation could exist on an EMS run, where a BLS unit would wait on the scene if the originally assigned paramedic unit was coming from five minutes away. However, with a critical patient, a hospital ten minutes away, and the paramedic unit coming from twenty minutes away, the EMTs may decide to transport immediately.
In all of these cases it comes down to taking the simple steps to ensure the most effective outcome. Verify assigned units are responding, take the correct action if they are not, and ensure that the people who need to know- KNOW. Dispatchers can play a very important role in ensuring effective and timely response of units, no matter what the emergency and its starts with verifying that response.