  In this work, we consider the two- and three-dimensional convective
Brinkman-Forchheimer (CBF) equations (or damped Navier--Stokes equations) on a
torus $\mathbb{T}^d,$ $d\in\{2,3\}$:
  $$ \frac{\partial \boldsymbol{u}}{\partial t}-\mu
\Delta\boldsymbol{u}+(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}+\alpha\boldsymbol{u}+\beta|\boldsymbol{u}|^{r-1}\boldsymbol{u}+\nabla
p=\boldsymbol{f}, \ \nabla\cdot\boldsymbol{u}=0,$$ where $\mu,\alpha,\beta>0$
and $r\in[1,\infty)$ is the absorption exponent. For $d=2,r\in[1,\infty)$ and
$d=3,r\in[3,\infty)$ ($2\beta\mu\geq 1$ for $d=r=3$), we first show the
backward uniqueness of deterministic CBF equations by exploiting the
logarithmic convexity property and the global solvability results available in
the literature. As a direct consequence of the backward uniqueness result, we
first derive the approximate controllability with respect to the initial data
(viewed as a start controller). Secondly, we apply the backward uniqueness
results in the attractor theory to show the zero Lipschitz deviation of the
global attractors for 2D and 3D CBF equations. By an application of
log-Lipschitz regularity, we prove the uniqueness of Lagrangian trajectories in
2D and 3D CBF flows and the continuity of Lagrangian trajectories with respect
to the Eulerian initial data. Finally, we consider the stochastic CBF equations
with a linear multiplicative Gaussian noise. For $d=2,r\in[1,\infty)$ and
$d=3,r\in[3,5]$ ($2\beta\mu\geq 1$ for $d=r=3$), we show the pathwise backward
uniqueness as well as approximate controllability via starter controller
results. In particular, the results obtained in this work hold true for 2D
Navier--Stokes equations.
