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The modelling of electrokinetic flows is a critical aspect spanning many industrial applications and research fields. This has introduced great demand in flexible numerical solvers to describe these flows. The underlying phenomena are microscopic, non-linear, and often involve multiple domains. Therefore often model assumptions and several numerical approximations are introduced to simplify the solution. In this work, we present a multi-domain multi-species electrokinetic flow model including complex interface and bulk reactions. After a dimensional analysis and an overview of some limiting regimes, we present a set of general purpose finite-volume solvers, based on \of, capable of describing an arbitrary number of electrochemical species over multiple interacting (solid or fluid) domains \cite{spnpfoam}. We provide verification of the computational approach for several cases involving electrokinetic flows, reactions between species, and complex geometries. We first present three one-dimensional verification test cases, and then show the capability of the solver to tackle two- and three-dimensional electrically driven flows and ionic transport in random porous structures. The purpose of this work is to lay the foundation for a general-purpose open-source flexible modelling tool for problems in electrochemistry and electrokinetics at different scales.