||Problems involving nonlinear time-dependent heat conduction in materials which have temperature-dependent thermal properties are solved with a novel meshless numerical solution technique using multiquadric radial basis functions (RBFs). Unlike traditional RBF collocation methods, the local Hermitian interpolation (LHI) method examined here can be scaled to arbitrarily large problems without numerical ill-conditioning or computational cost issues, due to the presence of small overlapping interpolation systems which grow in number but not in size as the global dataset grows. The flexibility of the full-domain multiquadric collocation method to directly interpolate arbitrary boundary conditions is maintained, via the local interpolations.The Kirchhoff transformation is employed to reduce the degree of nonlinearity in the governing PDE, and a high-resolution interpolation procedure is outlined to transform the various thermal properties to Kirchhoff-space. The implementation procedure is validated using a problem with analytical thermal properties and a known analytical solution. Additionally the procedure is validated against a problem with pointwise-measured material property data, and an analytical solution which is imposed via the body-source term. In this second case, the solution quality is compared with the traditional full-domain multiquadric collocation method.