Minimal model program for normal pairs along log canonical locus

Let $(X,\mathrm{\Delta })$ be a normal pair with a projective morphism $X\to Z$ and let $A$ be a relatively ample $\mathbb{R}$-divisor on $X$. We prove the termination of some minimal model program on $(X,\mathrm{\Delta }+A)\mathrm{/}Z$ and the abundance conjecture for its minimal model under assumptions that the non-nef locus of $K_X+\mathrm{\Delta }+A$ over $Z$ does not intersect the non-lc locus of $(X,\mathrm{\Delta })$ and that the restriction of $K_X+\mathrm{\Delta }+A$ to the non-lc locus of $(X,\mathrm{\Delta })$ is semi-ample over $Z$.