We develop a macroscopic description of the space-time evolution of the energy-momentum tensor during the pre-equilibrium stage of a high-energy heavy-ion collision. Based on a weak coupling effective kinetic description of the microscopic equilibration process (à la “bottom-up”), we calculate the nonequilibrium evolution of the local background energy-momentum tensor as well as the nonequilibrium linear response to transverse energy and momentum perturbations for realistic boost-invariant initial conditions for heavy-ion collisions. We demonstrate how this framework can be used on an event-by-event basis to propagate the energy-momentum tensor from far-from-equilibrium initial-state models to the time <math><msub><mi>τ</mi><mtext>hydro</mtext></msub></math> when the system is well described by relativistic viscous hydrodynamics. The subsequent hydrodynamic evolution becomes essentially independent of the hydrodynamic initialization time <math><msub><mi>τ</mi><mtext>hydro</mtext></msub></math> as long as <math><msub><mi>τ</mi><mtext>hydro</mtext></msub></math> is chosen in an appropriate range where both kinetic and hydrodynamic descriptions overlap. We find that for <math><mrow><msqrt><msub><mi>s</mi><mrow><mi>N</mi><mi>N</mi></mrow></msub></msqrt><mo>=</mo><mn>2.76</mn><mspace width="0.16em"></mspace><mspace width="0.16em"></mspace><mtext>TeV</mtext></mrow></math> central Pb-Pb collisions, the typical timescale when viscous hydrodynamics with shear viscosity over entropy ratio <math><mrow><mi>η</mi><mo>/</mo><mi>s</mi><mo>=</mo><mn>0.16</mn></mrow></math> becomes applicable is <math><mrow><msub><mi>τ</mi><mtext>hydro</mtext></msub><mo>∼</mo><mn>1</mn><mspace width="0.16em"></mspace><mrow><mi>fm</mi><mo>/</mo><mi>c</mi></mrow></mrow></math> after the collision.