The deformation dynamics of bottlebrush networks in a melt state is studied using a combination of theoretical, computational, and experimental tethniques. Three main molecular relaxation processes are identified in these systems: (i) relaxation of the side chains, (ii) relaxation of the bottlebrush backbones on length scales shorter than the bottlebrush Kuhn length (b(K)), and (iii) relaxation of the bottlebrush network strands between cross-links. The relaxation of side chains having a degree of polymerization (DP), n(sc), dominates the network dynamics on the time scales tau(0) < t <= tau(sc), where tau(0) and tau(sc) approximate to tau(0)(n(sc) + 1)(2) are the characteristic relaxation times of monomeric units and side chains, respectively. In this time interval, the shear modulus at small deformations decays with time as G(0)(BB)(t) similar to t(-1/2). On time scales t > tau(sc), bottlebrush elastomers behave as networks of filaments with a shear modulus G(0)(BB)(t) similar to (n(sc) + 1)(-1/4)t(-1/2). Finally, the response of the bottlebrush networks becomes time independent at times scales longer than the Rouse time of the bottlebrush network strands, tau(BB) similar to tau N-0(2)(n(sc) + 1)(3/2), where N is DP of the bottlebrush backbone between cross-links. In this time interval, the network shear modulus depends on the network molecular parameters as G(0)(BB)(t) similar to (n(sc) + 1)N--1(-1). Analysis of the simulation data shows that the stress evolution in the bottlebrush networks during constant strain-rate deformation can be described by a universal function. The developed scaling model is consistent with the dynamic response of a series of poly(dimethylsiloxane) bottlebrush networks (n(sc) = 14 and N = 50, 70, 100, 200) measured experimentally.