Under the assumption that $X$ is $\mathbb{Q}$-factorial, section 5 of the paper http://arxiv.org/pdf/math/0606666 addressed this issue, which was also proved earlier in Ambro's paper. Basically, if you assume the pair $(X,\Delta)$ is klt(lc), , so is the base $(Z,\Delta_Z)$ for some $\Delta_Z$. As the example of Prokhorov shows, this is optimal, namely, the base may not be terminal (canonical) even you assume the $(X,\Delta)$ is. For lc case, I think dlt modification+perturbation reduce the question to the klt case.
Section 5 of the paper http://arxiv.org/pdf/math/0606666 addressed this issue, which was also proved earlier in Ambro's paper. Basically, if you assume the pair $(X,\Delta)$ is klt (lc), so is the base $(Z,\Delta_Z)$ for some $\Delta_Z$, and \Delta_Z$. As the example of Prokhorov shows, this is optimal, namely, the base may not be terminal (canonical) even you assume the$(X,\Delta)$is. 1 The paper http://arxiv.org/pdf/math/0606666 addressed this issue, which was also proved earlier in Ambro's paper. Basically, if you assume the$(X,\Delta)$is klt (lc), so is the base$(Z,\Delta_Z)$for some$\Delta_Z$, and this is optimal, namely, the base may not be terminal (canonical) even you assume the$(X,\Delta)\$ is.