Ask Mr. Smarty Plants

We should begin by saying that Cotoneaster is native to China and the Himalayas and, as such, will not appear in our Native Plant Database. The Lady Bird Johnson Wildflower Center is dedicated to the growth, propagation and protection of plants native to North America. We can provide you with some links that may help you discover if your existing plants are poisonous to goats. We can also suggest an alternate way to get rid of those plants, since that is apparently your intent. Begin by cutting down the existing plants, as close to the soil as you can get. Within 5 minutes, using a disposable sponge paintbrush, brush that exposed trunk surface with a wide-spectrum herbicide. Do not use a spray, and be very careful not to spill or otherwise disperse the herbicide, as you will kill things you did not intend to do. Getting the herbicide on there quickly will allow the substance to enter the plant roots before they heal themselves over for protection. You can repeat this on every plant that is big enough to cut back. and will probably have to do it several times. As sprouts come up, clip or mow them back to the ground. Those sprouts provide leaves to continue to nourish the roots-no sprouts and the roots will eventually die, but not easily. If you don't already own hungry goats, we don't recommend you acquire any. 

Gardenweb Forum Why do I hate my cotoneasters?

Previous Mr. Smarty Plants answer on What do goats eat?

From another Mr. Smarty Plants answer:

"Whenever Mr. Smarty Plants gets questions about toxic plants, he checks out these databases to look for answers.

The Merck Veterinary Manual

ASPCA   

University of Arkansas 
 
University of Illinois  (common names only)    

Web site about Dobermans 

Toxic Plants of Texas

Poisonous Plants of North Carolina 

Cornell University Plants Poisonous to Livestock

University of Pennsylvania Poisonous Plants

Canadian Poisonous Plants Information System

California Poison Action Line

From the University of Arkansas website listed above on  Cotoneaster:

Toxicity Category 1 - Plants can cause systemic toxicity varying from mild abdominal cramping to serious cardiac complications.

Toxicity Category 3 - plants usually produce only mild to moderate stomach upset or dermatitis.