QUERÉTARO, Mexico—Jody Fledderman is one of five original employees at the auto-parts factory his father founded in rural Batesville, Ind. He also spends a lot of time at the company’s 97,000-square-foot plant in central Mexico.

The two operations have expanded together as automotive production in both countries boomed. Mr. Fledderman credits the success of Batesville Tool & Die Inc., where he is president, in part to the addition of a Mexican plant 16 years ago that helps service Honda Motor Co., Nissan Motor Co. and other...