A basepoint free theorem for algebraically integrable foliations

We show that if $\mathcal{F}$ is an algebraically integrable foliation on a $\mathbb{Q}$-factorial normal projective variety $X$, $A,B\ge 0$ are $\mathbb{Q}$-divisors on $X$ with $A$ ample such that $(\mathcal{F},B)$ is foliated dlt and $K_{\mathcal{F}}+A+B$ is nef, then $K_{\mathcal{F}}+A+B$ is semiample. We also provide some applications of this and related results such as contraction theorem for F-dlt pairs and a special case of the b-semiampleness conjecture.