  Let X be a product of Drinfeld modular curves over a general base ring A of
odd characteristic. We classify those subvarieties of X which contain a
Zariski-dense set of CM points. This is an analogue of the Andr\'e-Oort
conjecture. As an application, we construct non-trivial families of higher
Heegner points on modular elliptic curves over global function fields.
