Free electrons move around in a metal at random, colliding with eachother and with the positive ions in the metal.
When a metal rod is heated, the free electrons at the heated end gain kinetic energy.
The free electrons transfer this energy from the hotter end to the cooler end by moving faster, diffusing and colliding more with the cooler electrons and ions.
In a non-metallic solid, all the electrons are contained in the atoms.
In a non-metallic solid, conduction only happens due to vibrating atoms, which is much less effective than conduction in a metal.
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