A well-known case of evolutionary adaptation is that of ribulose-1,5-bisphosphate carboxylase (RubisCO), the enzyme responsible for fixation of CO2 during photosynthesis. Although the majority of plants use the ancestral C3 photosynthetic pathway, many flowering plants have evolved a derived pathway named C4 photosynthesis. The latter concentrates CO2, and C4 RubisCOs consequently have lower specificity for, and faster turnover of, CO2. The C4 forms result from convergent evolution in multiple clades, with substitutions at a small number of sites under positive selection. To understand the physical constraints on these evolutionary changes, we reconstructed in silico ancestral sequences and 3D structures of RubisCO from a large group of related C3 and C4 species. We were able to precisely track their past evolutionary trajectories, identify mutations on each branch of the phylogeny, and evaluate their stability effect. We show that RubisCO evolution has been constrained by stability-activity tradeoffs similar in character to those previously identified in laboratory-based experiments. The C4 properties require a subset of several ancestral destabilizing mutations, which from their location in the structure are inferred to mainly be involved in enhancing conformational flexibility of the open-closed transition in the catalytic cycle. These mutations are near, but not in, the active site or at intersubunit interfaces. The C3 to C4 transition is preceded by a sustained period in which stability of the enzyme is increased, creating the capacity to accept the functionally necessary destabilizing mutations, and is immediately followed by compensatory mutations that restore global stability.