Using irreversible thermodynamics we define and analyze dynamic limits for various traditional and work-assisted processes of sequential development with finite rates important in engineering. These dynamic limits are functions rather than numbers; they are expressed in terms of classical exergy change and a residual minimum of dissipated exergy, or some extensions including time penalty. We consider processes with heat and mass transfer that occur in a finite time and with equipment of finite dimension. These processes include heat-mechanical and (in Part 11) separation operations and are found in heat and mass exchangers, thermal networks, energy convertors, energy recovery units, storage systems, chemical reactors, and chemical plants. Our analysis is based on the condition that in order to make the results of thermodynamic analyses usable in engineering economics it is the dynamical limit, not the maximum of thermodynamic efficiency, which must be overcome for prescribed process requirements. A creative part of this paper outlines a general approach to the construction of "Carnot variables" as suitable controls. In this (first) part of work we restrict to dynamic limits on work that may be produced or consumed by a single resource flowing in an open heat-mechanical system. To evaluate these limits we consider sequential work-assisted unit operations, in particular those of heating or evaporation which run jointly with "endoreversible" thermal machines (e.g., heat pumps). We also compare structures of optimization criteria describing these limits. In particular, we display role of endoreversible limits in conventional operations of heat transfer and in work-assisted operations. Mathematical analogies between entropy production expressions in these two sorts of operations are helpful to formulate optimization criteria in both cases. Finite-rate models include minimal irreducible losses caused by thermal resistances to the classical exergy potential. Functions of extremum work, which incorporate residual minimum entropy production, are formulated in terms of initial and final states, total duration and (in discrete processes) number of stages.