We develop a general approach to the problem of the propagation dynamics of higher-order multipole spatial
solitons in nonlocal nonlinear media. By introducing the generalized Hermite-Laguerre-Gaussian ansatz similar to the corresponding linear modes, we construct nonlocal solitons by means of the variational approach and then find them numerically as stationary states. We then study the example of a tripole-like structure with nonzero angular momentum and investigate the stable propagation of such a beam in optical media with a nonlocal Gaussian response. Our results provide the first example of a self-trapped multi-vortex beam in self-focusing nonlinear media.