As has already been noted, the original electret microphone requires operating power (mistakenly called "bias" at times). This power should NOT be applied to a dynamic microphone. So you should either disconnect the (likely) resistor which feeds the power, or block the DC from the microphone with a capacitor. The voltage is typically quite low (~5V) A 0.5 or 1uF capacitor would be more than adequate for voice frequencies. And the polarity is fixed, so back-to-back capacitors are not required in this case.
HOWEVER. Unless you have an unusual dynamic microphone (not identified), it almost certainly has a LOWER signal output than a typical electret microphone capsule. So you face the real risk of not being to completely modulate the radio signal because your audio signal is too low. Since you did not identify your radio we don't know whether it possibly has a gain adjustment that might be able to compensate for the lower microphone signal. But it would not be surprising to find that an additional gain stage was required to boost the dynamic microphone signal level.