When say a gas stove is ignited, there is only a moment of flame, however, the gas will begin to burn. The gas continues to burn even after the initial flame is gone, so my question is how does the subsequent gas also combust despite the lack of fire? Wouldn't it just begin to leak gas because there is no more fire to combust it?
You don't need a 'flame' to light a fire. You need a fuel, oxygen and heat. Since the oxygen supply is constant, that leaves fuel and heat. The constant stream of gas allows the gas that is already on fire to heat the new gas, which continues as long as the gas is kept on.
I think your confusion stems from your misunderstanding of how combustion works. A 'flame' is simply the visual result of a chemical reaction. It is not one of the components needed for combustion