Leaf spots are a rare problem caused by one of several fungi. Use a spray of basic copper sulfate.
Cankers or twig blights are caused by fungi in the genera Physalospora, Diplodia and Gibberella. When a canker girdles the stem, the growth beyond the point of infection dies. There is no chemical control. Prune out the infected stems.
Cotoneasters are susceptible to fire blight, though C. adpressa is reportedly resistant. There is no chemical control for the disease. The disease kills the branch tips and the leaves appear scorched. Prune out infected twigs and avoid using high nitrogen fertilizer.