That is normal. Usually with the same amount of lattice constant change, the higher angle peaks shift more than the small angle ones. It is much easier to understand uising the Brag law.
Here La3+ replacing Y3+ and hence certain change in bond distance between Y-O and La-O that you know. Also it is clear that XRD pattern shows peaks with decreasing d spacing value and increasing 2theta value. Now let say at lower 2theta value, d-spacings are larger than at higher 2theta value. As a result relative change of d-spacing (that is change in d-spacing/actual d-spacing) would be negligible at larger d-spacing value, i.e, lower 2theta value and it would be significant at smaller d-spacing value, i.e, higher 2theta value, what you are observing in your present XRD pattern. I hope it makes some sense!
Have you measured the XRD at the same parameters for both samples.? It seems after La3+ treatment your sample turned amorphous and some doublet peaks are formed after doping too. In next experiment measure the XRD for better acquisition time to improve resolution etc. L2O3 and Y2O3 have major reflections at 2 theta of ca. 28.2 and 28.6 and peaks from these species are simply overlapped or doping is not done physically.. Study a little about the ionic radii of Y3+ and La3+ that will help you a bit to comment whether doping is truly possible. If yes is the case; study that doping by fixing the concentration of Y2O3 as a function of systematic concentration of La3+. Apply Vegard´s law at change in the peak position... etc that will help you to explain the shift.
It is the usual trend - the peaks at greater angles are shifted to a greater extent. You simply do not observe the shifts at low angles. La is greater than Y, thus the lattice parameter in the absece of any structural transofmation must increase on doping. However, all this depends on the concentration of solid solution according to Vegard's law.