In this paper, we consider comparison-based stochastic algorithms for solving numerical optimisation problems. We consider a specific subclass of algorithms called comparison-based step-size adaptive randomized search (CB-SARS), where the state variables at a given iteration are a vector of the search space and a positive parameter, the step-size, typically controlling the overall variance of the underlying search distribution. We investigate the linear convergence of CB-SARS on scaling-invariant objective functions. Scaling-invariant functions preserve the ordering of points with respect to their function value when the points are scaled with the same positive parameter (the scaling is done w.r.t. a fixed reference point). This class of functions includes norms composed with strictly increasing functions as well as non quasi-convex and non-continuous functions. On scaling-invariant functions, we show the existence of a homogeneous Markov Chain, as a consequence of natural invariance properties of CB-SARS (essentially scale-invariance and invariance to strictly increasing transformation of the objective function). We then derive sufficient conditions for asymptotic global linear convergence of CB-SARS, expressed in terms of different stability conditions of the normalised homogeneous Markov chain (irreducibility, positivity, Harris recurrence, geometric ergodicity) and thus define a general methodology for proving global linear convergence of CB-SARS algorithms on scaling-invariant functions.