The distinguishing variant ∆LIP of the Lattice Isomorphism Problem underlies the security proofs of a growing family of lattice-based schemes. As the special genus is the strongest efficiently computable invariant of a lattice, such a proof may replace the lattice used by the scheme with a random lattice of the same genus. This raises the question of how hard lattice problems actually are on such random lattices. We answer it, under GRH, with a worst-case to average-case reduction inside any special genus G of Hermitian lattices of fixed rank r ⩾ 2 over a cyclotomic field K of degree d: for each of (H)SVP, HCVP, (∆)BDD, DSP and DGS, solving the problem on a random class of G, drawn from its natural distribution, is as hard as solving it on the worst-case lattice of G, at the cost of a factor p = poly(d)·detQ(Λ)Or (1)/d in the approximation factor. The reduction is a random walk on the Kneser p-neighbour graph of G: neighbours stay close enough to transfer problem instances, while the walk equidistributes quickly, a fact we deduce from spectral bounds for the associated Hecke operators. As an application we give a direct and tight security proof for a minimal LIP-based KEM, resting on two independent assumptions: that the lattice it uses is indistinguishable from a random lattice of its genus, and that ∆BDD is hard on that genus. Unlike existing LIP-based proofs, this requires neither an ad-hoc lattice of the form qΛ ⊕ (q + 1)Λ, nor the tightness loss caused by the dense sublattice such proofs rely on.