The AV₃Sb₅ kagome superconductors series are of intense interest due to their diverse and intricate properties. The breaking of time-reversal symmetry (TRS) in the normal state of these superconductors stands as a significant feature, yet the extent to which this effect can be tuned remains uncertain. Here, we employ a unique low-energy muon spin rotation technique combined with local field numerical analysis to study the TRS breaking response as a function of depth from the surface in single crystals of RbV₃Sb₅ with charge order and Cs(V0.86Ta0.14)₃Sb₅ without charge order. In the bulk (specifically above 30 nm from the sur face) of RbV₃Sb₅, we have detected a notable increase in the internal field width experienced by the muon ensemble. This increase occurs within the charge ordered state. Intriguingly, the muon spin relaxation rate is significantly enhanced near the surface of RbV₃Sb₅ (specifically within a depth range of 30-40 nm from the surface), and this effect commences at temperatures significantly higher than the onset of charge order. Conversely, in Cs(V0.86Ta0.14)₃Sb₅, which lacks charge order, we do not observe a similar enhancement in the internal field width, neither in the bulk nor at the surface. These observations indicate a strong connection between charge order and TRS breaking on one hand, and on the other hand, suggest that TRS breaking can be manipulated to occur prior to long-range charge order. This research offers compelling evidence for depth-dependent magnetism in AV₃Sb₅ superconductors in the presence of charge order. Such findings are likely to illuminate the intricate microscopic mechanisms that underpin the TRS breaking phenomena in these materials.