We report structural, transport and magnetic measurements on the solid solutions of Ce 1− x La x NiSn and Ce 1− x La x RhSb ( x = 0 to 1). The results have been used to compare and contrast the effect of La substitution on the ground state properties of the Kondo insulators CeNiSn and CeRhSb. The resistivity measurements reveal that the energy gap in CeNiSn and CeRhSb decreases with increasing La concentration; no gap type behaviour was observed for La concentration ≥ 0.2. At low temperature, Ce 1− x La x NiSn alloys show a cross-over from a Kondo insulator ( x = 0) to a single ion Kondo ground state ( x ≥ 0.2) with increasing La concentration, while Ce 1− x La x RhSb alloys show a cross-over from a Kondo insulator ( x = 0) to a coherent Kondo lattice ( x = 0.2) and eventually to a single ion Kondo ground state at x = 0.5. Magnetic susceptibility measurements of Ce 1− x La x NiSn alloys exhibit a dramatic change in behaviour with La concentration; Curie-Weiss behaviour for x = 0, valence fluctuation behaviour for x = 0.2 and again CW behaviour for x ≥ 0.4. On the other hand, susceptibility of Ce 1− x La x RhSb alloys continue to show a valence fluctuation behaviour up to x = 0.7, and CW behaviour for higher La concentrations. CeNiSn and CeRhSb exhibit a large negative magnetoresistance at 20 T field and at 1.5 K.