This paper provides a systematic derivation of a guiding-center kinetic model that describes intense beam propagation through a periodic focusing lattice with axial periodicity length S, valid for sufficiently small phase advance (say, s , 60 ± ). The analysis assumes a thin ͑a, b ø S͒ axially continuous beam, or very long charge bunch, propagating in the z direction through a periodic focusing lattice with transverse focusing coefficients k x ͑s 1 S͒ k x ͑s͒ and k y ͑s 1 S͒ k y ͑s͒, where S const is the lattice period. By averaging over the (fast) oscillations occurring on the length scale of a lattice period S, the analysis leads to smooth-focusing Vlasov-Maxwell equations that describe the slow evolution of the guidingcenter distribution functionf b ͑x,ȳ,x 0 ,ȳ 0 , s͒ and (normalized) self-field potentialc͑x,ȳ, s͒ in the fourdimensional transverse phase space ͑x,ȳ,x 0 ,ȳ 0 ͒. In the resulting kinetic equation forf b ͑x,ȳ,x 0 ,ȳ 0 , s͒, the average effects of the applied focusing field are incorporated in constant focusing coefficients k x sf . 0 and k y sf . 0, and the model is readily accessible to direct analytical investigation. Similar smoothfocusing Vlasov-Maxwell descriptions are widely used in the accelerator physics literature, often without a systematic justification, and the present analysis is intended to place these models on a rigorous, yet physically intuitive, foundation.