The Ilkeston-based business has been sold by owner and managing director Bob Loach for an "undisclosed fee" to a management team headed up by operations director Nick Smith.
Following completion of the deal, Smith will become managing director while Drayton, previously production manager, will take on a director role at the business.
Loach becomes chairman of the 40 staff firm and retains a "modest" stake in the £1.5m turnover business.
He said: "Both Nick and Ann have been running the company for the last three years so I don't think it was sensible for me to stay in the capacity of managing director.
"They have decades of experience between them and I know that the business is in safe hands under their leadership."
FF Allsopp & Co runs a range of equipment that includes a 20 clamp Muller Martini Star Plus Binder, Wohlenberg Golf 18 Binder and Muller Martini 321 stitch line.
"This company has been going a long time and our customers have high expectations of us. We are known for both quality and reliability and that will not change," he added.