  We study adiabatic quantum quenches across a quantum multicritical point
(MCP) using a quenching scheme that enables the system to hit the MCP along
different paths. We show that the power-law scaling of the defect density with
the rate of driving depends non-trivially on the path, i.e., the exponent
varies continuously with the parameter $\alpha$ that defines the path, up to a
critical value $\alpha= \alpha_c$; on the other hand for $\alpha \geq
\alpha_c$, the scaling exponent saturates to a constant value. We show that
dynamically generated and {\it path($\alpha$)-dependent} effective critical
exponents associated with the quasicritical points lying close to the MCP (on
the ferromagnetic side), where the energy-gap is minimum, lead to this
continuously varying exponent. The scaling relations are established using the
integrable transverse XY spin chain and generalized to a MCP associated with a
$d$-dimensional quantum many-body systems (not reducible to two-level systems)
using adiabatic perturbation theory. We also calculate the effective {\it
path-dependent} dimensional shift $d_0(\alpha)$ (or the shift in center of the
impulse region) that appears in the scaling relation for special paths lying
entirely in the paramagnetic phase. Numerically obtained results are in good
agreement with analytical predictions.
