Does "had" here mean "had as a feature"?

And the feature is "with a dampened response to stress"?

Context:

In earlier research, Bale's lab had shown that male mice that were stressed, prior to being bred, by such means as changing cages or exposing them to a predator odor of fox urine, had offspring with a dampened response to stress. When they compared sperm from the stressed fathers to their unstressed counterparts, they found increased expression of nine miRs in the stressed animals. Unlike some other types of RNA, miRs do not code for a protein; instead, they serve to silence or degrade specific messenger RNAs, preventing them from being translated into proteins.

More:
http://www.sciencedaily.com/releases/2015/10/151019154111.htm