IMO, one must think of Consciousness and Name/Form not as entities but rather as mutually interdependent processes occurring within the inter linked non trivially recurrent cyclic process of Dependent Origination - they are fuelled by craving. Processes can cease, but they can also restart so long as there is fuel and the appropriate conditions reappear. Though the process of Consciousness runs on the nominal shell of what superficially appears to be a unique entity viz the body, it is not restricted to it… the body and its sense bases represent appropriate conditions true, but the body itself is not the fuel. Hence, even when this particular body is no longer operative, the processes of Consciousness and Name/Form do not go out of existence- the fuel of craving is not yet over. Instead, they re-establish on the basis of another suitable body (See the analogy of the spark going from one bonfire to another in SN44.9 posted by Christopher above).
Another way to understand the independence of the mind from the body, using modern analogies is by means of a thought experiment I described here.
