The tunneling-percolation mechanism of conduction in disordered conductor-insulator composites is studied for a realistic continuum model where conducting and impenetrable spherical particles are dispersed in a three-dimensional continuum insulating material. Conduction between particles is via tunneling processes and a maximum tunneling distance d is introduced. We determine the percolation critical concentration for several values of d. By doing so, we relax the restrictions applied in the previous studies of the problem, i.e., the considerations of the underlying lattice and the contribution of only the nearest neighbors. The tunneling-percolation transport is then analyzed by studying the conductance of the composite at and near the percolation threshold using a decimation procedure and a conjugate gradient algorithm. We show that at the critical concentration, and independently of the tunneling parameters, the critical transport exponent t reduces to the universal value t(0) similar or equal to 2, while moving away from the percolation threshold, the conductance exponent becomes larger than t(0), acquiring a strong concentration dependence. We interpret this feature as arising from the peculiar form of the distribution function for the local tunneling conductances. Consequently, apparent non-universality of transport appears when the conductance of the composite is fitted by forcing the exponent to be independent of the concentration. This leads us to believe that our tunneling-percolation theory is sufficient to explain the nonuniversal transport exponents observed in real disordered conductor- insulator compounds.