Does the chain continue if I keep knocking out the reinforcements

That's exactly how chains work, yes.

>>30089853
Here are the two things that will cause a chain to end;
- summoner has a major status effect
- new summoner is no longer the same evolutionary family

>>30089959
>- summoner has a major status effect
what?

>>30090174
If you paralyze/poison/burn(probably haven't tested myself) your summoner of allies, and there are no more allies left, the chain ends. I had this happen accidentally as I had a Pikachu out, the rockruff used tackle and paralyzed itself via contact, and then it wouldn't summon any more rockruffs

Does the chain break when no one shows up?

>>30089959
Status effects don't end a chain, they just prevent it from continueing while the status is ongoing.

>>30092950
No. ONLY summons when there are no more Pokémon of the same evolution line as the original Pokémon encountered.

That is the ONLY thing that actually breaks the chain. Other factors can contribute to that happening sooner, but they don't actually break the chain themselves.