The High-Lu­mi­nos­ity LHC (HL-LHC) crab cav­i­ties (CCs) will be in­stalled on both sides of IP1 (ATLAS) and IP5 (CMS) to com­pen­sate for the geo­met­ric lu­mi­nos­ity re­duc­tion due to the cross­ing angle. To cope with the in­creased beam cur­rent (0.55 A DC for LHC, 1.1 A for HL-LHC), the op­er­a­tion of the LLRF sys­tem has been changed: rather than fully com­pen­sat­ing the tran­sient beam load­ing, we allow the phase to vary along the turn (100 ps peak-peak with 1.1 A DC). This has been im­ple­mented at LHC since July 2017. The CCs have high loaded Q (5e5) and the avail­able RF power is in­suf­fi­cient to fol­low the bunch phase mod­u­la­tion. The crab­bing volt­age is not mod­u­lated, caus­ing a phase error w.r.t. the in­di­vid­ual bunch cen­troids, lead­ing to trans­verse kicks of the cen­troids and an asym­met­ric crab­bing of the bunch cores. We pre­sent an an­a­lyt­i­cal model for the re­sult­ing lu­mi­nos­ity re­duc­tion and val­i­date with par­ti­cle track­ing sim­u­la­tions. Due to the sym­me­try of the bunch fill­ing pat­terns for the counter-ro­tat­ing beams, the peak lu­mi­nos­ity is re­duced by only 2% for nom­i­nal HL-LHC pa­ra­me­ters at IPs 1 and 5, which is within tol­er­a­ble lim­its.