A many-body theory of paramagnetic Kondo insulators is described, focusing specifically on single-particle dynamics, scattering rates, dc transport and optical conductivities. This is achieved by development of a non-perturbative local moment approach to the symmetric periodic Anderson model within the framework of dynamical mean-field theory. Our natural focus is the strong-coupling, Kondo lattice regime, in particular the resultant `universal' scaling behaviour in terms of the single, exponentially small low-energy scale characteristic of the problem. Dynamics/transport on all relevant (omega, T)-scales are considered, from the gapped/activated behaviour characteristic of the low-temperature insulator through to explicit connection to single-impurity physics at high omega and/or T; and for optical conductivities emphasis is given to the nature of the optical gap, the temperature scale responsible for its destruction and the consequent clear distinction between indirect and direct gap scales. Using scaling, explicit comparison is also made to experimental results for dc transport and optical conductivities of Ce(3)Bi(4)Pt(3), SmB(6) and YbB(12). Good agreement is found, even quantitatively; and a mutually consistent picture of transport and optics results.