We have investigated the glass-transition kinetics of nine Ge_xSe_1−x glasses by differential scanning calorimetry. Relation between the drive (heating-rate q) and response (heatflow shift ΔH at T_g) is seen to be strictly linear only for GeSe₄, known to signify the bulk-rigidity threshold for this series. From an Arrhenius analysis the activation energies for glassy relaxation are estimated, and point to the existence of different thermokinetic phases below and above the threshold composition. Series behaviour of the kinetic activation is conciled to a concurring one seen in the size of cooperatively diffusing regions. The anomalies are attributed to structural crossovers with Ge doping; first from the parent uniform Se-chains to that of backbones out-branching at corner-shared Ge(Se_1⁄2)₄ tetrahedral clusters, and subsequently interconnecting by edge-shared configurations to realize a random pearl-necklace 3-D covalent network.