Folding transformations of equations from the Gambier family

•Using a generalized folding transformation we establish a connection between the generalized second-order Abel and Gambier equations.•As an application (dynamical) of folding transformation we find that the Darboux polynomials are invariant.•In consequence of this invariance the first integrals are also invariant. Using Okamoto’s folding transformation we investigate the mapping of the Gambier equation and its higher-order analogs to the generalized Abel chain of equations. In particular we show how the Darboux polynomials and first integral of the Abel equation can be mapped to an equation of the Gambier type using folding transformations.