In this work we focus on a group of Galactic double neutron star ( DNS ) systems with long orbital periods of \gtrsim 1 day and low eccentricities of \lesssim 0.4 . The feature of these orbital parameters is used to constrain the evolutionary processes of progenitor binaries and the supernova ( SN ) kicks of the second born NSs . Adopting that the mass transfer during primordial binary evolution is highly non-conservative ( rotation-dependent ) , the formation of DNS systems involves a double helium star binary phase , the common envelope ( CE ) evolution initiates before the first NS formation . During the CE evolution the binary orbital energy is obviously larger when using a helium star rather than a NS to expel the donor envelope , this can help explain the formation of DNS systems with long periods . SN kicks at NS birth can lead to eccentric orbits and even the disruption of binary systems , the low eccentricities require that the DNSs receive a small natal kick at the second collapse . Compared with the overall distribution of orbital parameters for observed DNS binaries , we propose that the second born NSs in most DNS systems are subject to small natal kicks with the Maxwellian dispersion velocity of less than 80 km s ^ { -1 } , which can provide some constraints on the SN explosion processes . The mass distribution of DNS binaries is also briefly discussed . We suggest that the rotation-dependent mass transfer mode and our results about SN kicks should be applied to massive binary evolution and population synthesis studies .