  We demonstrate that multiple higher-order topological transitions can be
triggered via the continuous change of the geometry in kagome photonic crystals
composed of three dielectric rods. By tuning a single geometry parameter, the
photonic corner and edge states emerge or disappear with the higher-order
topological transitions. Two distinct higher-order topological insulator phases
and a normal insulator phase are revealed. Their topological indices are
obtained from symmetry representations. A photonic analog of fractional corner
charge is introduced to distinguish the two higher-order topological insulator
phases. Our predictions can be readily realized and verified in configurable
dielectric photonic crystals.
