We explore the far-from-equilibrium dynamics of a (0+1)-dimensionally expanding system with Bjorken symmetry using kinetic theory and hydrodynamics. For a conformal scenario, i.e., when the underlying medium is composed of massless excitations, the shear Reynolds number in kinetic theory is known to exhibit an early-time attractor, which is well described by higher-order hydrodynamics. In sharp contrast, we show that the breaking of conformal invariance by inclusion of even a small mass of particles can drastically alter this attractor behaviour, with neither the shear nor the bulk Reynolds numbers depicting early-time universality. However, the scaled effective longitudinal pressure continues to show fast decay to an attractor in non-conformal kinetic theory, driven by the rapid expansion of Bjorken flow at early times. We shall discuss how this finding sheds new light on the previously observed attractors in weakly-coupled conformal systems. We shall also present the formulation of a hydrodynamic theory which describes the non-conformal kinetic theory attractor with excellent precision.