Hybrid elastic and spin waves hold promises for energy-efficient and versatile generation and detection of magnetic signals, with potentially long coherence times. Here we report on the combined elastic and magnetic dynamics in a one-dimensional magnetomechanical crystal composed of an array of magnetic nanostripes. Phononic and magnonic modes are impulsively excited by an optical ultrafast trigger and their decay is monitored by time-resolved magneto-optical Kerr effect. Complementary Brillouin light scattering measurements and micromagnetic simulations concur in a unified picture, in which the strength and degree of mixing of coherent and dissipative coupling of the quasiparticles are determined quantitatively.