The D and $D_s$ decay constants are important inputs for the determination of the CKM matrix elements $|V_{cd}|$ and $|V_{cs}|$ and for precision tests of the Standard Model. With lattice determinations of these quantities reaching the sub-percent level, it is important to demonstrate control over all sources of systematic uncertainty, and, in particular, those arising from discretisation effects and the quark mass dependence. We present results obtained from a large number of $N_f=2+1$ CLS ensembles (including two ensembles at the physical point) which span 6 lattice spacings in the range $0.04 \le a \le 0.1$ fm and lie on three trajectories in the quark mass plane.