Near extinction transient behaviors of CH4/N2 versus O2/N2 counterflow nonpremixed flames were investigated by adopting a porous spherical burner with a large radius of curvature to achieve a very low buoyant stretch rate. By varying the oxygen concentrations in the oxidizer stream, four regimes were identified by three limits of sooting, instability, and extinction. The sooting limit was sensitive to flame temperature, and was characterized by two indexes of fuel injection velocity uF and oxygen concentration XO2. When decreasing uF, the sooting limit was different from high stretch flames because of appreciable heat loss toward the burner surface. In the instability regime, periodic holes were observed whose frequency decreased with XO2. Periodic holes were also found to occur with N2 dilution in the fuel stream, which were more regular with larger frequencies compared with the periodic holes in the diluted oxidizer streams. The edge propagation behaviors for periodic holes in diluted fuel streams were analyzed. The statistical average of the propagation speed of advancing edge (hole closing) Se decreased with increasing fuel concentration gradient, and the ratio of Se with the stoichiometric laminar burning velocity SLst of Se/SLst was larger than unity. Also, the stretch rate significantly affected the propagation speed of advancing edge flames. Determination of the propagation speed of the retreating edge (hole opening) had intrinsic difficulty in estimating the transient radial flow velocity, however, the approximated |Se/SLst| was larger than unity and decreased with the fuel mole fraction. Nitrogen dilution in the oxidizer stream was proven to be more effective than in the fuel stream for flame suppression of methane. This study could enrich the understanding of very low-stretch nonpremixed flames in a counterflow burner and provide information regarding low-stretch flame suppression, which may be relevant to microgravity flame behaviors.