Volumes of foliations birationally bounded by algebraically integrable families

We prove that for log canonical foliations which are birationally bounded by algebraically integrable families, the set of their volumes satisfies the DCC. This answers a special case of a question posed by Cascini, Hacon, and Langer. As a key ingredient, we establish the deformation invariance of relative log canonical volumes for a family of weak semistable morphisms, which can be viewed as a relative version of the classical result proved by Hacon, McKernan, and Xu.