Convection is a form of heat transfer (as well as conduction and radiation). Convection is based on mass/density differences between substances near and away from a heat source. Items near the heat source will rise in temperature. Thus, their molecules will gain kinetic energy and expand. This results in a lower density. Items with a lower density rise above those with a higher density, so the heated molecules rise upwards. As the molecules move away from the heat source, they will lose kinetic energy and contract. This results in a greater density. At this point, the molecules begin to sink. Hence, a cycle of hot molecules rising and cool molecules sinking begins. This cycle is called a convection current.
Thus, the ash of the burning piece of paper in your question will rise and then fall back down again.