  We study the transport properties of a quantum wire, described by the
Tomonaga-Luttinger model, in the presence of a backscattering potential
provided by several extended time-dependent impurities (barriers). Employing
the B\" uttiker-Landauer approach, we first consider the scattering of
noninteracting electrons ($g=1$) by a rectangular-like barrier and find an
exact solution for the backscattering current, as well as a perturbative
solution for a weak static potential with an arbitrary shape. We then include
electron-electron interactions and use the Keldysh formalism combined with the
bosonization technique to study oscillating extended barriers. We show that the
backscattering current off time-dependent impurities can be expressed in terms
of the current for the corresponding static barrier. Then we determine the
backscattering current for a static extended potential, which, in the limit of
noninteracting electrons ($g=1$), coincides with the result obtained using the
B\" uttiker-Landauer formalism. In particular, we find that the conductance can
be increased beyond its quantized value in the whole range of repulsive
interactions $0<g<1$ already in the case of a single oscillating extended
impurity, in contrast %contrary to the case of a point-like impurity, where
this phenomenon occurs only for $0<g<1/2$.
