This paper deals with global asymptotic behaviour of the dynamics for $N$-dimensional type-K competitive Kolmogorov systems of differential equations defined in the first orthant. It is known that the backward dynamics of such systems is type-K monotone. Assuming the system is dissipative and the origin is a repeller, it is proved that there exists a compact invariant set $Sigma$ which separates the basin of repulsion of the origin and the basin of repulsion of infinity and attracts all the non-trivial orbits. There are two closed sets $S_H$ and $S_V$, their restriction to the interior of the first orthant are $(N-1)$-dimensional hypersurfaces, such that the asymptotic dynamics of the type-K system in the first orthant can be described by a system on either $S_H$ or $S_V$: each trajectory in the interior of the first orthant is asymptotic to one in $S_H$ and one in $S_V$. Geometric and asymptotic features of the global attractor $Sigma$ are investigated. It is proved that the partition $Sigma = Sigma_HcupSigma_0cupSigma_V$ holds such that $Sigma_HcupSigma_0subset S_H$ and $Sigma_VcupSigma_0subset S_V$. Thus, $Sigma_0$ contains all the $omega$-limit sets for all interior trajectories of any type-K subsystems and the closure $overline{Sigma_HcupSigma_V}$ as a subset of $Sigma$ is invariant and the upper boundary of the basin of repulsion of the origin. This $Sigma$ has the same asymptotic feature as the modified carrying simplex for a competitive system: every nontrivial trajectory below $Sigma$ is asymptotic to one in $Sigma$ and the $omega$-limit set is in $Sigma$ for every other nontrivial trajectory.