Let $\pi:X \to U$ be a projective morphism, and $(X, \Delta = A + B)$ be a KLT pair, where $A$ is a general ample divisor and $B$ is effective.

Suppose $K_X + \Delta$ is not nef (over $U$) and there exists a nef divisor $C$ such that $K_X + \Delta + C$ is nef. Then there exists an extremal ray $R$ which is $(K_X + \Delta)$-negative and there exists $\lambda \in (0,1]$ such that $K_X + \Delta + \lambda C$ is nef but trivial on $R$. Now, we run MMP with scaling $C$, my questions are the following:

Suppose $f: X \to Z$ is the contraction of the extremal ray $R$, if it is a divisoral contraction, in order to run MMP, we need to use $(Z, \Delta')$ replace $(X, \Delta)$ where $\Delta' = f_* \Delta$, why in this case $K_Z + \Delta' + \lambda C'$ is nef ( here $C'$ is $f_*C$)?

Similarly, if $f$ is a small contraction, and suppose the flip $X^+$ exists, why $K_{X^+}+ \Delta^+ + \lambda C^+$ is nef?