We propose higher-order isoparametric finite element approximations for mean curvature flow and surface diffusion. The methods are natural extensions of the piecewise linear finite element methods introduced by Barrett, Garcke, and Nürnberg (BGN) in a series of papers in 2007 and 2008. The proposed schemes exhibit unconditional energy stability and inherit the favorable mesh quality of the original BGN methods. Moreover, in the case of surface diffusion we present structure-preserving higher-order isoparametric finite element methods. In addition to being unconditionally stable, these also conserve the enclosed volume. Extensive numerical results demonstrate the higher-order spatial accuracy, the unconditional energy stability, the volume preservation for surface diffusion, and the good mesh quality.