Radial age gradients hold the cumulative record for the multitude of physical processes driving the build-up of stellar populations and the ensuing star formation (SF) quenching process in galaxy bulges and, therefore, potentially sensitive discriminators between competing theoretical concepts on bulge formation and evolution. Based on spectral modeling of integral field spectroscopy (IFS) data from the CALIFA survey, we derived mass- and light-weighted stellar age gradients (∇(t⋆, B)ℒ, ℳ) within the photometrically determined bulge radius (RB) of a representative sample of local face-on late-type galaxies that span 2.6 dex in stellar mass (8.9 ≤ log ℳ⋆, T ≤ 11.5). Our analysis documents a trend of decreasing ∇(t⋆, B)ℒ, ℳ with increasing ℳ⋆, T, with high-mass bulges predominantly showing negative age gradients and vice versa. The inversion from positive to negative ∇(t⋆, B)ℒ, ℳ occurs at log ℳ⋆, T ≃ 10, which roughly coincides with the transition from lower-mass bulges whose gas excitation is powered by SF to bulges classified as composite, LINER, or Seyfert. We discuss two simple limiting cases for the origin of radial age gradients in massive late-type galaxy bulges. The first one assumes that the stellar age in the bulge is initially spatially uniform (∇(t⋆, B)ℒ, ℳ ≈ 0), thus the observed age gradients (∼ - 3 Gyr/RB) arise from an inside-out SF quenching (ioSFQ) front that is radially expanding with a mean velocity vq. In this case, the age gradients for massive bulges translate into a slow (vq ∼1-2 km s-1) ioSFQ that lasts until z ∼ 2, suggesting mild negative feedback by SF or an active galactic nucleus (AGN). If, on the other hand, negative age gradients in massive bulges are not due to ioSFQ but primarily due to their inside-out formation process, then the standard hypothesis of quasi-monolithic bulge formation has to be discarded in favor of another scenario. This would involve a gradual buildup of stellar mass over 2-3 Gyr through, for instance, inside-out SF and inward migration of SF clumps from the disk. In this case, rapid (≪1 Gyr) AGN-driven ioSFQ cannot be ruled out. While the ℳ⋆, T versus ∇(t⋆, B)ℒ, ℳ relation suggests that the assembly history of bulges is primarily regulated by galaxy mass, its large scatter (∼1.7 Gyr/RB) reflects a considerable diversity. This calls for an in-depth examination of the role of various processes (e.g., negative and positive AGN feedback, bar-driven gas inflows) with higher-quality IFS data in conjunction with advanced spectral modeling codes.