Gamma-ray bursts (GRBs) have been phenomenologically divided into long-and short-duration populations, generally corresponding to the collapsar and compact merger origin, respectively. Here we collect three unique bursts, GRBs 060614, 211211A and 211227A, all characterized by a long-duration main emission (ME) phase and a rebrightening extended emission (EE) phase, to study their observed properties and the potential origin as neutron star-black hole (NSBH) mergers. NS-first-born (BHfirst-born) NSBH mergers tend to contain fast-spinning (non-spinning) BHs that more easily (hardly) allow tidal disruption to happen with (without) forming electromagnetic signals. We find that NS-firstborn NSBH mergers can well interpret the origins of these three GRBs, supported by that: (1) Their X-ray MEs and EEs show unambiguous fall-back accretion signatures, decreasing as ∝ t −5/3 , which might account for their long duration. The EEs can result from the fall-back accretion of r-process heating materials, predicted to occur after NSBH mergers. (2) The beaming-corrected local event rate density for this type of merger-origin long-duration GRBs is R 0 ∼ 2.4 +2.3 −1.3 Gpc −3 yr −1 , consistent with that of NS-first-born NSBH mergers. (3) Our detailed analysis on the EE, afterglow and kilonova of the recently high-impact event GRB 211211A reveals it could be a merger between a ∼ 1.23 +0.06 −0.07 M NS and a ∼ 8.21 +0.77 −0.75 M BH with an aligned-spin of χ BH ∼ 0.62 +0.06 −0.07 , supporting an NS-first-born NSBH formation channel. Long-duration burst with rebrightening fall-back accretion signature after ME, and bright kilonova might be commonly observed features for on-axis NSBH mergers. We estimate the multimessenger detection rate between gravitational waves, GRB and kilonova emissions from NSBH mergers in O4 (O5) is ∼ 0.1 yr −1 (∼ 1 yr −1 ).