  We consider general radially moving frames realized in the background of
nonextremal black holes having causal structure similar to that of the
Schwarzschild metric. In doing so, we generalize the Lema\^{\i}tre approach,
constructing free-falling frames which are built from the reference particles
with an arbitrary specific energy $e_{0}$ including $e_{0}<0$ and a special
case $e_{0}=0$. The general formula of 3-velocity of a freely falling particle
with the specific energy $e$ with respect to a frame with $% e_{0}$ is
presented. We point out the relation between the properties of considered
frames near a horizon and the Banados-Silk-West effect of an indefinite growth
of energy of particle collisions. Using our radially moving frames, we consider
also nonradial motion of test particles including the regions near the horizon
and singularity. We also point out the properties of the Lema\^{\i}tre time at
horizons depending on the frame and sign of particle energy.
