We consider the multigraded Hilbert scheme corresponding to the Hilbert function of a finite number of points in general position in a smooth projective complex toric variety. We develop several criteria for a point of that parameter space to be in the distinguished irreducible component. To obtain the criteria we study the behaviour of the locus of saturated ideals under morphisms of multigraded Hilbert schemes. We apply our results to classify the irreducible multigraded Hilbert schemes corresponding to points in general position in a product of projective spaces.