General radially moving references frames in the black hole background

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 $$ 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.