We study the broad emission lines in a sample of 11 gravitationally lensed quasars with at least two epochs of observation to identify intrinsic variability and to disentangle it from microlensing. To improve our statistical significance and emphasize trends, we also include 15 lens systems with single-epoch spectra. Mg II and C III] emission lines are only weakly affected by microlensing, but C IV shows strong microlensing in some cases, even for regions of the line core, presumably associated with small projected velocities. However, excluding the strongly microlensed cases, there is a strikingly good match, on average, between the red wings of the C IV and C III] profiles. Analysis of these results supports the existence of two regions in the broad-line region (BLR), one that is insensitive to microlensing (of size ≳50 lt-day and kinematics not confined to a plane) and another that shows up only when it is magnified by microlensing (of size of a few light-days, comparable to the accretion disk). Both regions can contribute in different proportions to the emission lines of different species and, within each line profile, to different velocity bins, all of which complicates detailed studies of the BLR based on microlensing size estimates. The strength of the microlensing indicates that some spectral features that make up the pseudo-continuum, such as the shelf-like feature at λ1610 or several Fe III blends, may in part arise from an inner region of the accretion disk. In the case of Fe II, microlensing is strong in some blends but not in others. This opens up interesting possibilities to study quasar accretion disk kinematics. Intrinsic variability seems to affect the same features prone to microlensing, with similar frequency and amplitude, but does not induce outstanding profile asymmetries. We measure intrinsic variability (≲20%) of the wings with respect to the cores in the C IV, C III], and Mg II lines consistent with reverberation mapping studies.