The notion of topological (Thouless) pumping in topological phases is traditionally associated with Laughlin's pump argument for the quantization of the Hall conductance in two-dimensional (2D) quantum Hall systems. It relies on magnetic flux variations that thread the system of interest without penetrating its bulk, in the spirit of Aharonov-Bohm effects. Here we explore a different paradigm for topological pumping induced, instead, by magnetic flux variations δχ inserted through the bulk of topological phases. We show that δχ generically controls the analog of a topological pump, accompanied by robust physical phenomena. We demonstrate this concept of bulk pumping in two paradigmatic types of 2D topological phases: integer and fractional quantum Hall systems and topological superconductors. We show that bulk pumping provides a unifying connection between seemingly distinct physical effects such as density variations described by Streda's formula in quantum Hall phases and fractional Josephson currents in topological superconductors. More importantly, we argue that bulk pumping provides a generic tool for probing topological phases and inducing robust physical effects, similar in spirit yet crucially different from Laughlin's pump. We discuss its generalizations in other topological phases.