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We consider the gradient flow of the Yang–Mills–Higgs functional of Higgs pairs on a Hermitian vector bundle $(E,H_0)$ over a Kähler surface $(M,\omega )$, and study the asymptotic behavior of the heat flow for Higgs pairs at infinity. The main result is that the gradient flow with initial condition $(A_0,{\phi }_0)$ converges, in an appropriate sense which takes into account bubbling phenomena, to a critical point $(A_{\infty },{\phi }_{\infty })$ of this functional. We also prove that the limiting Higgs pair $(A_{\infty },{\phi }_{\infty })$ can be extended smoothly to a vector bundle $E_{\infty }$ over...