Homotopy relative Rota-Baxter Lie algebras, triangular $L_infty$-bialgebras and higher derived brackets

We describe $L_infty$-algebras governing homotopy relative Rota-Baxter Lie algebras and triangular $L_infty$-bialgebras, and establish a map between them. Our formulas are based on a functorial approach to Voronovs higher derived brackets construction which is of independent interest.