While looking for information about quasifibrations I stumbled over a E-Mail of Tom Goodwillie (https://www.lehigh.edu/~dmd1/tg516.txt) where he introduces different definitions of quasifibrations and some Lemmas corresponding to that. He defines the notion of being a universal quasifibration so that it should fulfil that every pullback of a weak equivalence along a universal quasifibration is a weak equivalence.
Definition UQF: A map $p$ is a universal quasifibration (UQF) if $p$ itself and every pullback of this map is a quasifibration.
Then he gives an easier way to check if something is a UQF.
The original text:
Lemma 2: If p becomes QF under every base change to a closed
disk, then p is UQF.
[Remark: A map to a contractible space is QF iff the inclusion of each fiber into the total space is a weak equivalence.]
Proof (easy, sketched): We have only to show that p is QF. This means showing that certain extension/lifting problems w.r.t p have solutions. Every such problem really "lives" over some disk that's mapped to B, so it's enough if we can solve it after pulling back to the disk.
I do understand that the remark is true and I also understand why it is enough to show that $p$ is a quasifibration but I do not know how to do that. Especially I do not know how being a quasifibration is connected to extension/lifting problems w.r.t. $p$.