The generalization analysis of deep kernel learning (DKL) is a crucial and open problem of kernel methods for deep learning. The implicit nonlinear mapping in DKL makes existing methods of capacity-based generalization analysis for deep learning invalid. In an attempt to overcome this challenge and make up for the gap in the generalization theory of DKL, we develop an analysis method based on the composite relationship of function classes and derive capacity-based bounds with mild dependence on the depth, which generalizes learning theory bounds to deep kernels and serves as theoretical guarantees for the generalization of DKL. In this paper, we prove novel and nearly-tight generalization bounds based on the uniform covering number and the Rademacher chaos complexity for deep (multiple) kernel machines. In addition, for some common classes, we estimate their uniform covering numbers and Rademacher chaos complexities by bounding their pseudo-dimensions and kernel pseudo-dimensions, respectively. The mild bounds without strong assumptions partially explain the good generalization ability of deep learning combined with kernel methods.