Try using Contact between the plate hole ID and the bushing OD and use Contact Step Control in single, multi-step Transient Thermal analysis.

Step 1, Contact is disabled so no heat is conducted from the plate to the bushing.  Heat up the plate using some value for Internal Heat Generation.

Step 2, Set the Internal Heat Generation to 0 and wait for the plate to cool off.

Step 3, Contact is enabled and heat can begin to conduct into the bearing, wait till the End Time and examine the temperature in both parts.

I expect you are using a Convection BC to allow the plate to cool, you would also want that on the bushing. If the Ambient temperature on the Convection BC is the same as the Environment temperature that the bushing picks up at the start of the simulation, the bushing should behave the way you want.