<p>This paper is concerned with transition fronts of bistable reaction–diffusion systems in domains with multiple cylindrical branches. We first show that there is a unique front-like entire solution originating from planar fronts in some branches; then, under the complete propagation we show that the front-like entire solution is a transition front and converges to planar fronts (with some shift) in other branches. Here an entire solution is referred to a solution that is defined for all time <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(t\in \mathbb {R}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation> and in the whole space <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(x\in \Omega \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>∈</mo> <mi mathvariant="normal">Ω</mi> </mrow> </math></EquationSource> </InlineEquation>. Furthermore, we prove the existence and uniqueness of the global mean speed of any transition front, assuming the complete propagation of the front-like entire solutions emanating from each of the branches. Finally, we give some sufficient conditions on the domain to ensure that the propagation is complete.</p>