A novel design of hollow-core Bragg fibers (HC-BFs) with linearly-chirping distributed dielectric cladding is proposed, for demand of the multi-component trace-gas detection based on the mid-infrared (mid-IR) absorption spectrum. The photonic band-gap (PBG) structures and transmission-loss characteristics for the TE and TM modes under nearly glancing incidence condition in such a quasi-periodic HC-BF are numerically investigated and compared with a conventional one with periodic cladding. The results show that the quasi-periodic cladding can effectively enlarge the PBG width by increasing the linear increment of cladding period and dielectric-layer number. The enlargement effect enhances with the increment of both structure parameters. Meanwhile, the central wavelength of the PBG shifts to the longer wavelength. In the enlarged wavelength range of PBG, the HC-BF with linearly-chirping distributed dielectric cladding still has the low loss, in the order of 0.01 dB/m, showing the excellent performance of mid-IR low-loss broadband transmission.