This paper presents a method to find new De Bruijn cycles based on ones of lesser order. This is done by mapping a De Bruijn cycle to several vertex disjoint cycles in a De Bruijn digraph of higher order and connecting these cycles into one full cycle. We characterize homomorphisms between De Bruijn digraphs of different orders that allow this construction. These maps generalize the well-known D-morphism of Lempel between De Bruijn digraphs of consecutive orders. Also, an efficient recursive algorithm that yields an exponential number of nonbinary De Bruijn cycles is implemented.