We will present a complete set of equations, in the form of an Einstein-Bianchi system, that describe the evolution of generic smooth lattices in spacetime. All 20 independent Riemann curvatures will be evolved in parallel with the leg lengths of the lattice. We will show that the evolution equations for the curvatures forms a hyperbolic system and that the associated constraints are preserved. This work is a generalization of our previous paper [L. Brewin, Phys. Rev. D 85, 124045 (2012)] on the Einstein-Bianchi system for the Schwarzschild spacetime to general 3 + 1 vacuum spacetimes.