We consider processes consisting of a category of states varying over a control category as prescribed by a unique factorisation lifting functor. After a brief analysis of the structure of general processes in this setting, we restrict attention to linearly-controlled ones. To this end, we introduce and study a notion of path-linearisable category in which any two paths of morphisms with equal composites can be linearised (or interleaved) in a canonical fashion. Our main result is that categories of linearly-controlled processes (viz., processes controlled by path-linearisable categories) are sheaf models.