A convergence theorem is obtained for quantum random walks with particles in an arbitrary normal state. This unifies and extends previous work on repeated-interactions models, including that of Attal and Pautrat (2006, Ann. Henri Poincaré 7, 59-104) and the author (2010, J. London Math. Soc. (2) 81, 412-434; 2010, Comm. Math. Phys. 300, 317–329). When the walk generator acts by ampliation and either multiplication or conjugation by a unitary operator, necessary and sufficient conditions are given for the quantum stochastic cocycle which arises in the limit to be driven by an isometric, co-isometric or unitary process.