  We study the measurement of Higgs boson self-couplings through $2\rightarrow
3$ vector boson scattering (VBS) processes in the framework of Standard Model
effective field theory (SMEFT) at both proton and lepton colliders. The SMEFT
contribution to the amplitude of the $2\to3$ VBS processes, taking $W_L
W_L\rightarrow W_L W_L h$ and $W_L W_L\rightarrow h h h$ as examples, exhibits
enhancement with the energy
$\frac{\mathcal{A}^{\text{BSM}}}{\mathcal{A}^{\text{SM}}} \sim
\frac{E^2}{\Lambda^2}$, which indicates the sensitivity of these processes to
the related dimension-six operators in SMEFT. Simulation of the full processes
at both hadron and lepton colliders with a variety of collision energies are
performed to estimate the allowed region on $c_6$ and $c_{\Phi_1}$. Especially
we find that, with the help of exclusively choosing longitudinal polarizations
in the final states and suitable $p_T$ cuts, $WWh$ process is as important as
the more widely studied triple Higgs production ($hhh$) in the measurement of
Higgs self-couplings. Our analysis indicates that these processes can play
important roles in the measurement of Higgs self-couplings at future 100 TeV pp
colliders and muon colliders. However, their cross sections are generally tiny
at low energy machines, which makes them much more challenging to explore.
