Fourier’s law fails when the mean free path of the energy carriers becomes comparable to the length and time scales over which the temperature field varies. We derive a thermodynamically consistent generalization in which the conductivity becomes a nonlocal memory operator ${𝛋}_{eff}(𝐤,\omega )$, by combining mesoscopic nonequilibrium thermodynamics with the Mori–Kubo–Zwanzig projection-operator formalism. The Onsager kernel is a tensor sum over vibrational normal modes weighted by their Bose heat capacities and relaxation functions; it satisfies the second law for any closure consistent with the fluctuation–dissipation theorem and reduces, in the relaxation-time closure, to the nonlocal conductivity of the phonon Boltzmann equation. Two consequences follow. First, direction-dependent sampling of this tensor in a pump–probe experiment makes a nominally isotropic crystal exhibit apparent conductivities ${\Lambda }_z\ne {\Lambda }_r$. Second, the modulation frequency does not introduce temporal memory but sets the probed wavevector through the thermal penetration depth, so that the frequency-dependent suppression of the apparent conductivity measured by time-domain thermoreflectance on Ge and SiGe alloys is a spatial-nonlocality effect, with effective lengths of $0.25$–$0.4\mu$m. With published mean-free-path spectra and no parameter adjusted to the thermoreflectance data, the kernel reproduces the spot-size dependence of Si, the frequency dependence of the apparent conductivity of Ge and Si${}_{0.2}$Ge${}_{0.8}$ and the rise of the apparent interface conductance of Al/Si${}_{0.99}$Ge${}_{0.01}$, and contains the two-channel ballistic/diffusive model as a two-sector projection.