  The opening of a spin gap in the orthorhombic compounds CeT$_2$Al$_{10}$ (T =
Ru and Os) is followed by antiferromagnetic ordering at $T_N$ = 27 K and 28.5
K, respectively, with a small ordered moment (0.29$-$0.34$\mu_B$) along the
$c-$axis, which is not an easy axis of the crystal field (CEF). In order to
investigate how the moment direction and the spin gap energy change with 10\%
La doping in Ce$_{1-x}$La$_x$T$_2$Al$_{10}$ (T = Ru and Os) and also to
understand the microscopic nature of the magnetic ground state, we here report
on magnetic, transport, and thermal properties, neutron diffraction (ND) and
inelastic neutron scattering (INS) investigations on these compounds. Our INS
study reveals the persistence of spin gaps of 7 meV and 10 meV in the 10\%
La-doped T = Ru and Os compounds, respectively. More interestingly our ND study
shows a very small ordered moment of 0.18 $\mu_B$ along the $b-$axis (moment
direction changed compared with the undoped compound), in
Ce$_{0.9}$La$_{0.1}$Ru$_2$Al$_{10}$, however a moment of 0.23 $\mu_B$ still
along the $c-$axis in Ce$_{0.9}$La$_{0.1}$Os$_2$Al$_{10}$. This contrasting
behavior can be explained by a different degree of hybridization in
CeRu$_2$Al$_{10}$ and CeOs$_2$Al$_{10}$, being stronger in the latter than in
the former. Muon spin rotation ($\mu$SR) studies on
Ce$_{1-x}$La$_x$Ru$_2$Al$_{10}$ ($x$ = 0, 0.3, 0.5 and 0.7), reveal the
presence of coherent frequency oscillations indicating a long$-$range
magnetically ordered ground state for $x$ = 0 to 0.5, but an almost temperature
independent Kubo$-$Toyabe response between 45 mK and 4 K for $x$ = 0.7. We will
compare the results of the present investigations with those reported on the
electron and hole$-$doping in CeT$_2$Al$_{10}$.
