In SGA 4.5 (Arcata V.1) Deligne writes:

Let $X$ be a complex analytic variety and $f:X\rightarrow D$ map $X$ into the disk. Write $[0,t]$ for closed line segment with extremities 0 and $t$ in $D$, and $]0,t]$ for the half open segment. If $f$ is smooth (lisse) the inclusion $j:f^{-1}(]0,t])\rightarrow f^{-1}([0,t])$ is a homotopy equivalence; one can push the special fiber $X_0=f^{-1}(0)$ into $f^{-1}([0,t])$.

I think that last $[0,t]$ at the end of the sentence is a typo and should say $]0,t]$. Is that right?

Of course this motivating example from analytic geometry is not itself exactly scheme theory but the comparison is clear. Anyway Deligne concludes this passage with: "one can express this construction imagistically by saying, for a smooth map, the general fiber swallows (avale) the special fiber." He uses the same image to motivate proper base change by comparison with point set topology.

I think the idea of the image, in scheme terms, is that the fiber at any point always maps to the fiber at any specialization -- but in general the special fiber does not map back to the generalization. And when it does map back he says the general fiber swallows the special fiber. Am I right to think of it that way?