  The Enskog kinetic equation is considered to determine the mobility $\lambda$
and diffusion $D$ transport coefficients of intruders immersed in a granular
gas of inelastic hard spheres (grains). Intruders and grains are in contact
with a thermal bath, which plays the role of a background gas. As usual, the
influence of the latter on the dynamics of intruders and grains is accounted
for via a viscous drag force plus a stochastic Langevin-like term proportional
to the background temperature $T_\text{b}$. The transport coefficients
$\lambda$ and $D$ are determined by solving the kinetic equation by means of
the Chapman--Enskog method adapted to dissipative dynamics. Both transport
coefficients are given in terms of the solutions of two integral equations
which are approximately solved up to the second order in a Sonine polynomial
expansion. Theoretical results are compared against numerical solutions of the
inelastic Enskog equation by means of the direct simulation Monte Carlo (DSMC)
method. Good agreement between theory and simulations is in general found,
specially in the case of the second Sonine approximation. The knowledge of the
coefficients $\lambda$ and $D$ allow us to assess the departure of the Einstein
relation $\epsilon=D/(T_{\text{b}}\lambda)$ from 1. As expected from previous
results for driven granular gases, it is shown that the origin of the deviation
of $\epsilon$ from 1 is only due to the non-Maxwellian behavior of reference
state of intruders (measured by the cumulant $c_0$) when the bath temperature
$T_\text{b}$ is replaced by the intruder temperature $T_0$ in the Einstein
relation. Since the magnitude of $c_0$ is in general very small, deviations of
the (modified) Einstein relation $\epsilon_0=D/(T_0\lambda)$ from 1 cannot be
detected in computer simulations of dilute granular gases. This conclusion
agrees well with previous computer simulation results.
