Fast detection of transient temperature change.. RTD? Thermistors? Thermocouples? or something else?
Short answer: it depends.  The response time has more to do with thermal mass and thermal conductivity, and little to do with electrical principles of thermocouples, thermistors, RTDs.  

If a sensor (of either type) has large thermal mass, it will take longer to equilibrate the temperature with the medium that it's measuring.  If a sensor is insulated from the medium, then it will have a slower response.  A sensor in water or oil will have a faster response than in air.

For example a thermocouple made with hair-thin wires will respond very quickly.  An industrial ungrounded thermoucouple probe will be slower.

edit:


  looking to track rate of combustion


I've heard that thin exposed thermocouples are used in the automotive industry for this type of measurement.  I've also heard that such thermocouples don't last long because combustion gases corrode them.  Having said that, I don't know in which part of the engine this was done.