Motivated by recent claims of a possible redshift dependence in late-Universe determinations of the Hubble constant (H₀), we test the robustness of this behaviour using multiple cosmological probes. We perform a joint redshift-binned analysis of H₀ across eight bins using late-Universe probes -- Pantheon+ SNe~Ia, DESI BAO, cosmic chronometers, and water megamasers -- under three cosmological frameworks: flat ΛCDM, CPL, and Padé cosmography. Under a common baseline scheme, all three models show a qualitatively similar, low-amplitude variation in the per-bin H₀ estimates. A simple Fourier-like parametrization captures this behaviour, but the amplitude differs from zero only at a marginal significance of about 1. 71--1. 94\, σ, with similar behaviour observed across all three cosmological frameworks. We then investigate the robustness and possible origin of this feature. Alternative binning schemes preserve its qualitative form, whereas single-probe per-bin fits (SNe-only, CC-only, BAO-only) yield ratios H₀, ₈/H₀, ₆₋₎₁₀₋ mostly consistent with unity and do not reproduce the pronounced drift seen in the joint baseline constraints. Finally, by comparing different global versus piecewise-constant configurations for \H₀, Ωₘ, M, rd\, we find that a baseline-like oscillatory pattern re-emerges only when multiple degenerate parameter combinations are allowed to vary across bins, while it is strongly suppressed when only H₀ is bin-dependent. Taken together, these results indicate that the apparent oscillatory behaviour of H₀ (z) in late-time arises from known parameter degeneracies and does not constitute robust evidence for a genuine redshift evolution.
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