We consider the balayage of a measure μ defined on a domain Ω onto its boundary ∂Ω. Assuming that Ω has a corner of opening πα at a point z₀ ∈ ∂Ω for some 0 < α ≤ 2 and that dμ(z) ≅ |z - z₀|²ᵇ⁻² dz² as z → z₀ for some b > 0, we obtain the precise rate of vanishing of the balayage of μ near z₀. The rate of vanishing is universal in the sense that it only depends on α and b. We also treat the case when the domain has multiple corners at the same point. Moreover, when 2b ≤ 1 / α, we provide explicit constants for the upper and lower bounds.