Well, as Cinth said, counter-waves meet your description perfectly, but if Counter-Posts isn't enabled, that shouldn't be it.
Warp Relays might do it, but that's also a plot that needs to be enabled.
Warp Guardians can send waves to inner systems if the timing is right, but those should be normal waves (appearing at the wormhole).
You can get some funky results sometimes if you conquer a system and leave a Troop Accelerator or similar sort of spawner in the system, but again - that's not normally counter-wave-like.
Showdown can do it, which is technically not a plot but a minor faction. I think you'd be aware of that one, though.
Marauders spawn like that, but normally don't appear without a battle already occurring.
Dreyshock, if you're willing, you can post your game here as an attachment, and I'm sure someone will be willing to take a look at it to see.