When in proxy-SVARs the covariance matrix of VAR disturbances is subject to exogenous, permanent breaks that cause IRFs to change across volatility regimes, even strong, exogenous external instruments yield inconsistent estimates of the dynamic causal effects. However, if these volatility shifts are properly incorporated into the analysis through (testable) "stability restrictions", we demonstrate that the target IRFs are point-identified and can be estimated consistently under a necessary and sufficient rank condition. If the shifts in volatility are sufficiently informative, standard asymptotic inference remains valid even with (i) local-to-zero covariance between the proxies and the instrumented structural shocks, and (ii) potential failures of instrument exogeneity. Intuitively, shifts in volatility act similarly to strong instruments that are correlated with both the target and non-target shocks. We illustrate the effectiveness of our approach by revisiting a seminal fiscal proxy-SVAR for the US economy. We detect a sharp change in the size of the tax multiplier when the narrative tax instrument is complemented with the decline in unconditional volatility observed during the transition from the Great Inflation to the Great Moderation. The narrative tax instrument contributes to identify the tax shock in both regimes, despite our empirical analysis raises concerns about its "statistical" validity.