  We have studied the transport coefficients as a tool to probe the collision
integral appeared in the Boltzmann equation. For this purpose, we have
estimated the transport coefficients (momentum: \{$\eta$,$\zeta$\}, heat:
\{$\kappa$\}, and charge: \{$\sigma_{\rm el}$\}) in the kinetic theory with
Bhatnagar-Gross-Krook (BGK) and the collision integral in RTA and ask whether
we can distinguish between the two collision integrals. For example, $\eta$
gets enhanced while $\zeta$ gets reduced w.r.t. to RTA. As a corollary, we then
investigate the interplay among the aforesaid transport coefficients, {\em
viz.} fluidity and transition point of QCD medium by evaluating the ratios,
$\eta/s$ and $\zeta/s$, respectively, nature of flow (Reynolds number, RI),
sound attenuation (Prandtl number, Pr), and competition between the momentum
and charge diffusion ($\gamma$) etc. as further plausible tools to decipher the
same. With BGK collision integral, the ratios, $\eta/s$ (increase) and
$\zeta/s$ (decrease) shows opposite behavior whereas Pr, RI, $\gamma$ and the
ratio $\zeta/\eta$ get reduced w.r.t RTA. We then examine how a strong $B$
modulate the impact of collision integral, which, in a way, explore the
dimensionality dependence of the momentum transport because the quark dynamics
gets restricted to 1-D only and only the lowest Landau levels are populated. As
a result, $\eta$ ($\zeta$) gets reduced (amplified), which will have
ramifications on the ratios, {\em viz.} $\eta/s$ ($\zeta/s$) becomes smaller
(larger), enhancement of Pr, $\gamma$ and $\zeta/\eta$ etc. In this study, the
thermo-magnetic medium effects have been incorporated by adopting a
quasi-particle model, where the medium generated masses of the partons have
been obtained from the pole of their resummed propagators calculated using
perturbative thermal QCD in strong $B$.
