A common feature of topological insulators is that they are characterized by topologically invariant quantity such as the Chern number and the $mathbb{Z}_2$ index. This quantity distinguishes a nontrivial topological system from a trivial one. A topological phase transition may occur when there are two topologically distinct phases, and it is usually defined by a gap closing point where the topologically invariant quantity is ill-defined. In this paper, we show that the magnon bands in the strained (distorted) kagome-lattice ferromagnets realize an example of a topological magnon phase transition in the realistic parameter regime of the system. When spin-orbit coupling (SOC) is neglected (i.e. no Dzyaloshinskii-Moriya interaction), we show that all three magnon branches are dispersive with no flat band, and there exists a critical point where tilted Dirac and semi-Dirac point coexist in the magnon spectra. The critical point separates two gapless magnon phases as opposed to the usual phase transition. Upon the inclusion of SOC, we realize a topological magnon phase transition point at the critical strain $delta_c=frac{1}{2}big[ 1-(D/J)^2big]$, where $D$ and $J$ denote the perturbative SOC and the Heisenberg spin exchange interaction respectively. It separates two distinct topological magnon phases with different Chern numbers for $delta<delta_c$ and for $delta>delta_c$. The associated anomalous thermal Hall conductivity develops an abrupt change at $delta_c$, due to the divergence of the Berry curvature in momentum space. The proposed topological magnon phase transition is experimentally feasible by applying external perturbations such as uniaxial strain or pressure.
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