Variable-coefficient nonlinear Schrödinger (NLS)-type models are used to describe certain phenomena in plasma physics, nonlinear optics, arterial mechanics, and Bose–Einstein condensation. In this article, the coupled variable-coefficient cubic-quintic NLS equations with external potentials in the non-Kerr fibre are investigated. Via symbolic computation, similarity transformations and relevant constraints on the coefficient functions are obtained. Based on those transformations, such equations are transformed into the coupled cubic-quintic NLS equations with constant coefficients. Nonautonomous soliton solutions are derived, and propagation and interaction of the nonautonomous solitons in the non-Kerr fibre are investigated analytically and graphically. Those soliton solutions are related to the group velocity dispersion r ( x ) and external potentials h 1 ( x ) and h 2 ( x , t ). With the different choices of r ( x ), parabolic, cubic, and periodically oscillating solitons are obtained; with the different choices of h 2 ( x , t ), we can see the dromion-like structures and nonautonomous solitons with smooth step-like oscillator frequency profiles, to name a few.