There are two basic types of radio trigger. Those that handshake/link, and those that don't.
The second type, those that are the most basic, are simply a transmitter that sends a signal, and a receiver that is looking for a signal. These basic models from Phottix, Yongnuo, Pocket Wizard etc, will work regardless of the order things are turned on. They can even be triggered by transmitters from different brands, as long as the signal is the right frequency.
The first type, which is becoming more common, i.e. the Phottix Odin, Pocket Wizard Flex, Broncolor RF etc, work using custom communication protocols that allows the transmitter to control functions on the receiver, adjusting power, working in E-TTL etc.
Some of these systems, those that are really well engineered, work regardless of the order things are turned on. e.g. the Phottix Odin.
Some of them do care, some more than others. The Pocket Wizard Flex can be very fussy unless things are turned on in the right order. The Broncolor RF sometimes requires the lights to be turned on before the trigger is woken from sleep mode, or it sometimes doesn't see all the lights.