The transfers of parameterized results along a special case $s \le t$ ($Propagation$) ask that, on every word, the parameter of an instance of $t$ be at most a function of the parameter of an instance of $s$. A parameter agrees along $s \le t$ if, on every word, the two parameters are equal; then it is bounded with the identity.

Theorem. Let $s \le t$ be specifications of the operations family, both with the standard encoding. Under the standard reading the parameters $n$, $m$, $p_{\max}$, $\#p$, $\#r$, $\#w$, $\#d$, $w_{\max}$, the number of kinds of machines, the rank of $(p_{ij})$, $q_{\max}$, $\chi$, the largest communication delay, the width of the precedence order, $pw(I)$ (the most time windows sharing an instant, minus one) and the maximum slack agree along $s \le t$, and so do their sums. The same holds for $n$, $p_{\max}$, $\#p$, $\#r$, $\#w$, $\#d$ and $w_{\max}$ in the batching family.

The words determine the kernel instance, the views and the threshold, and the lift changes none of them; every parameter listed reads only those. The time window of a job ends at its deadline, and also at $D$ for the makespan and at $d_j + D$ for the maximum lateness; the makespan is a special case only of itself and of the maximum lateness with all due dates $0$, so the windows agree.

Fixed-parameter tractability in the sense of lax-496464 includes that the parameter is computed in polynomial time. For a parameter that agrees along $s \le t$, the parameter of a word of $s$ is that of the same word read for $t$, so it is computed by the same program.

Theorem. If the reading lifts along $s \le t$ and determines the answer for $t$, then fixed-parameter tractability goes down from $t$ to $s$ for every parameter that agrees along $s \le t$.