The initial-boundary value problem of flexural dynamics of flat-reinforced elastic-plastic plates is formulated under von Kàrmàn’s assumptions. The structural elements of the composite are assumed to satisfy the plastic yielding theory with isotropic hardening. The obtained equations and the corresponding boundary conditions allow to compute the stress-strain state of flexible composite plates accounting for weak transverse shear resistance with various orders of accuracy. The proposed higher-order plate model gives the known relations of the Reddy theory for flexible plates as a first approximation. The time-step integration using centered difference approximations allows to construct an explicit leap-frog scheme in the case of blast-loaded plate. The elastic-plastic flexural dynamics of oblong rectangular and annular plates reinforced along principal strain directions is investigated under air wave loadings. The plate boundaries are clamped, and the rigid disks are rigidly connected with contours of annular plates’ holes. It is shown that the dynamic response computed on the background of the Reddy model differs significantly from the response given by refined plate theories at time moments upward of several tenths of second. Moreover the leap-frog difference schema based on refined plate dynamics equations offers better stability in practical computations as compared to Reddy’s plate theory.