If the temperature is plotted against length, it will fall more rapidly for wood than for metal. You may have come across an electrified fence in the country which is used to confine cattle to prescribed grazing areas and wondered if the wire is active or not and safe to touch. One way of doing this is to select a long, thin piece of dry grass, one end of which you hold and the other you place on the wire. If the fence is active you feel a tingling which will increase as you move the contact point of the grass with the wire closer to your hand. Make sure the grass is thin and not wet otherwise you may get a shock.
Conduction is the result of collisions between the particles making up the material.
The particles have some motion as a result of there being at some temperature and therefore have kinetic energy and momentum. During a collision energy and momentum can be exchanged and therefore transferred from one particle to another. Wherever a temperature gradient exists a net flow of heat will result. Imagine a number of stationary goods trucks which are coupled to each other and constrained to movement in one dimension by the railway lines on which they can roll. Applying an impulsive force to the truck at one end causes it to hit the adjacent truck and bounce back as a result of the buffers imparting some energy and momentum as it does so.