Given that different tissues require different amounts of blood at different times how do these specific tissues control their local blood flow?

The answer is not absolute but there are two theories that could explain how this happens.

Lack of oxygen theory.

Lets remember that for muscular contraction to happen oxygen is needed, so the whole process of the myosin bridges can be completed, so this theory establishes that due to a lack of oxygen the muscle fibers in the capillaries fail to contract producing an expansion of the vessels that in turn results in an increased blood flow.

This means that tissues that consume too much oxygen leave to little of this one available for the muscular contraction, which causes the above explained, ensuring thus an increased blood flow to a tissue that uses a lot of oxygen.

Vasodilator theory.

This theory says that the tissues more active will have an increased metabolism, this means that the production of vasodilator substances (which are produced during the activity of the tissue) will also be increased. This substances will act upon the capillaries causing their expansion which leads to an increased blood flood. So the more active tissues will have an increased blood flow.

The vasodilator substances produced are:

Adenosine.

CO2.

Histamine.

Potassium ions.

Hydrogen ions.

Source: Guyton and Hall Textbook of Medical Physiology 13th Ed.