  We study $(1+1)$-dimensional integrable soliton equations with time-dependent
defects located at $x=c(t)$, where $c(t)$ is a function of class $C^1$. We
define the defect condition as a B\"{a}cklund transformation evaluated at
$x=c(t)$ in space rather than over the full line. We show that such a defect
condition does not spoil the integrability of the system. We also study soliton
solutions that can meet the defect for the system. An interesting discovery is
that the defect system admits peaked soliton solutions.
