Research into the evolution of grammar requires that we employ formalisms and processing mechanisms that are powerful enough to handle features found in human natural languages. But the formalism needs to have some additional properties compared to those used in other linguistics research that are specifically relevant for handling the emergence and progressive co-ordination of grammars in a population of agents. This document introduces Fluid Construction Grammar, a formalism with associated parsing, production, and learning processes designed for language evolution research. The present paper focuses on a formal definition of the unification and merging algorithms used in Fluid Construction Grammar. The complexity and soundness of the algorithms and their relation to unification in logic programming and other unification-based grammar formalisms are discussed.