We analyse the magnon spectrum and distribution function of the antiferromagnetic phase of the Floquet-driven Hubbard model. Above a critical drive strength, we find a dynamical instability, resulting not from a drive-induced population inversion but rather from a change in sign of the magnon damping at a non-zero wavevector, in other words from a dynamically induced gain mechanism. The change in sign means that infinitesimal fluctuations grow with time, corresponding to an instability of the driven state. Implications for the nonequilibrium distribution function and the strong drive nonlinear dynamics are discussed. We analyse the magnon spectrum and distribution function of the antiferromagnetic phase of the Floquet-driven Hubbard model. Above a critical drive strength, we find a dynamical instability, resulting from a dynamically induced gain mechanism (change in sign in magnon damping at a non-zero wavevector) rather than from a drive-induced population inversion. Implications for the nonequilibrium distribution function and the strong drive nonlinear dynamics are discussed.