As time elapses after exercise, a person's pulse rate should decrease until his/her resting pulse rate is reached.
Energy is produced in the mitochondria of your cells during a process called cellular respiration. During cellular respiration, oxygen gas and the sugar glucose are used to produce carbon dioxide gas and water.
During exercise, your body uses energy at a faster rate. Thus, your body needs to deliver the reactants to your cells and pump out waste at a faster rate. During exercise, breathing increases so more oxygen is brought into your body and carbon dioxide is pumped out at a faster rate than normal.
Oxygen is carried to where it is needed by the hemoglobin in your red blood cells. During exercise, your heart needs to pump more blood in order to deliver enough oxygen to your working cells. Therefore, your heart begins to beat faster and with more force per beat.
As an individual stops exercising, not as much oxygen or glucose is needed to produce energy via cellular respiration. Therefore, one's heart rate and respiratory rate begin to slow down.
The link below provides a lab in which you can investigate the effects of time after exercise on pulse rate.