Why should jet engine throttles be moved smoothly?
The issue is the possibility of a compressor stall.  The combustion will directly respond to changes in the thrust lever but it takes time for this to spool up the turbine that is connected to the compressor.  If the incoming air from the compressor is not of adequate pressure because it hasn't spooled up enough you risk a compressor surge.  In these engines you need to slowly increase thrust so that the turbine can spool up (and hence the compressor) so the airflow through the engine remains adequate for the engine power.

Modern engines with FADEC systems should not have these issues as the FADEC can enforce rate limits for fuel flow increases or directly calculate the maximum safe fuel flow from observed engine parameters.  Modern engines might also have more sophisticated compressors with variable stators that can help mitigate compressor stalls/surges.