= Solution
Put $U=B_0-A_0$, $T=D_0-C_0$, $D=A_0-C_0$ and $v=V-W$. A point on each moving segment has the form $A_0+sU+tV$ and $C_0+rT+tW$, with $s,r\in[0,1]$. Contact is therefore equivalent to
$$
D+sU-rT+tv=0,\qquad0\le s,r\le1,\quad t\ge0.
$$
The <contact time of translating line segments> is the minimum feasible $t$. This reduces the entire continuous-motion problem to three <linear equations> and linear inequalities.
In three-dimensional space form the <matrix> $H=[U,-T,v]$. If it is nonsingular, solve
$$
\begin{pmatrix}s\\r\\t\end{pmatrix}=-H^{-1}D
$$
using a stable linear solver. A candidate with $s,r$ in the unit interval and $t\ge0$ is the only contact event and hence the earliest one; otherwise no contact occurs. Check initial intersection explicitly, so an already touching pair returns time zero.
If $H$ is singular, use a rank-revealing factorization to test consistency. Write every solution as $x=x_0+Nz$, with columns of $N$ spanning the <null space>, and minimize its time coordinate subject to the box and nonnegative-time constraints. This is a small <linear programming> problem. It handles parallel segments, collinear overlap, a continuum of contact times and segments degenerated to points. Infeasibility means no future contact. When $v=0$, the relative geometry is stationary: initially disjoint segments never meet. \b[The first contact is the least feasible nonnegative time, not just an intersection of the two supporting lines.]
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