Given a smooth globally hyperbolic spacetime M satisfying the Einstein vacuum equations and a timelike geodesic C in M, I will describe the construction of a family of metrics g(ε) which are close to g away from C, and which near C describe a Kerr black hole of mass ε traveling along C. These metrics are formal solutions of the Einstein field equations in that they satisfy the field equations modulo O(ε^N) for all N. In the special case that M is itself a subextremal Kerr or Kerr-de Sitter black hole spacetime and C crosses the future event horizon, we can thus describe extreme mass ratio mergers to all orders in perturbation theory.