The spatial derivatives in Hamilton-Jacobi partial differential equations for the definition of morphological operations such as dilation and erosion for grey-value images are replaced by fractional derivatives of arbitrary positive order. Focus is laid on geometric invariance with respect to reflections and rotations so that directional bias towards the coordinate directions is avoided. Discretisation of directional fractional derivatives via truncated general power series ultimately leads to an optimisation problem for the advection direction. We numerically compare the proposed fractional morphological operations with conventional counterparts and a simpler fractional-order alternative on grey-value images to show interesting phenomena and gain insights into the effects of the non-local nature of the fractional derivatives which merit further investigation.