  If two particles move towards a black hole and collide in the vicinity of the
horizon, under certain conditions their energy $E_{c.m.}$ in the center of mass
frame can grow unbounded. This is the Ba\~{n}ados-Silk-West (BSW) effect.
Usually, this effect is considered for extremal horizons and geodesic (or
electrogedesic) trajectories. We study this effect in a more general context,
when both geometric and dynamic factors are taken into account. We consider
generic axially symmetric rotating black holes. The near-horizon behavior of
metric coefficients is determined by three numbers $p,~q,$ $k$ that appear in
the Taylor expansions for different types of a horizon$.$ This includes
nonextremal, extremal and ultraextremal horizons. We also give general
classification of possible trajectories that include so-called usual,
subcritical, critical and ultracritical ones depending on the near-horizon
behavior of the radial component of the four-velocity. We assume that particles
move not freely but under the action of some unspecified force. We find when
the finiteness of a force and the BSW effect are compatible with each other.
The BSW effect implies that one of two particles has fine-tuned parameters. We
show that such a particle always requires an infinite proper time for reaching
the horizon. Otherwise, either a force becomes infinite or a horizon fails to
be regular. This realizes the so-called principle of kinematic censorship that
forbids literally infinite $E_{c.m.}$ in any act of collision. The obtained
general results are illustrated for the Kerr-Newman-(anti-)de Sitter metric
used as an example. The description of diversity of trajectories suggested in
our work can be of use also in other contexts, beyond the BSW effect. In
particular, we find the relation between a force and the type of a trajectory.
