We present a framework to address the dynamic response shaping question of nonlinear transport-reaction chemical processes. The spatiotemporal behavior of such processes can be described in the form of dissipative partial differential equations (PDEs), the modal infinite-dimensional representation of which can in principle be partitioned into two subsystems; a finite-dimensional slow and its complement infinite-dimensional fast and stable subsystem. The dynamic shaping problem is addressed via regulation of error dynamics between the process and a desired spatiotemporal behavior presented by a target PDE system. We approximate the infinite-dimensional nature of the systems via model order reduction; adaptive proper orthogonal decomposition (APOD) is used to compute and recursively update the set of empirical basis functions required by Galerkin projection to build switching reduced order models of the spatiotemporal dynamics. Then, the nonlinear output feedback control design is formulated by combination of a feedback control law and a nonlinear Luenberger type dynamic observer to regulate the error dynamics. The effectiveness of the proposed control approach is demonstrated on shaping the thermal dynamics of an exothermic reaction in a catalytic chemical reactor.