By using a high-intensity flying focus laser, the dephasingless [Phys. Rev. Lett. 124 134802 (2020)] or phase-locked [Nat. Photon. 14 475 (2020)] laser wakefield acceleration (LWFA) can be realized, which may overcome issues of laser diffraction, pump depletion, and electron dephasing which are always suffered in usual LWFA. The scheme thus has the potentiality to accelerate electrons to TeV energy in a single acceleration stage. However, the controlled electron injection has not been self-consistently included in such schemes. Only external injection was suggested in previous theoretical studies, which requires other accelerators and is relatively difficulty to operate. Here, we numerically study the actively controlled density transition injection in phase-locked LWFA to get appropriate density profiles for amount of electron injection. The study shows that compared with LWFA driven by lasers with fixed focus, a larger plasma density gradient is necessary. Electrons experience both transverse and longitudinal loss during acceleration due to the superluminal group velocity of the driver and the variation of the wakefield structure. Furthermore, the periodic deformation and fracture of the flying focus laser in the high-density plasma plateau make the final injected charge also depend on the beginning position of the density downramp. Our studies show a possible way for amount of electron injection in LWFA driven by flying focus lasers.