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$_{10}$.
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