What are the advantages of higher order Kalman Filters like EKF, UKF?
They should improve the convergence properties of the state estimate, i.e. if you have an initial value far from the trajectory, they could improve the region of attraction. They should also reduce the bias in the estimate.
In the EKF and UKF, there are no direct ways of including higher order moments of distributions, in both cases the state distributions are assumed to be Gaussian. Presumably, you can approximate non-Gaussian distributions with additional modeling. For (some) higher order approximations, it should be possible to include higher order moments. [However, as far as I understand, the additional computational complexity is usually not worth the performance gain (if any), since there are so many sources of uncertainty when modeling a system.] The go-to solution for determining the distributions of state-estimates is Monte-Carlo simulation.