It won't. Armory simply pushes the tx to its supporting node, the node propagates it to the rest of the network. There is no P2P command to dismiss a transaction from the mempool, that would be a massive DoS vector. In other words, once the tx is in the wild, you can only really hope it gets mined, if you aren't going to replace it.

Armory will not drop unconfirmed transactions from its DB until they either mine or are replaced. As the user, you can force Armory to wipe a 0 confirmation tx from your db. It won't dismiss it from mempools, as this can't be done.

Clearing up the zeroconf DB in Armory let's you pick the output for spending again, but there is no guarantee the new tx will propagate since it double spends the first one. Armory does not create RBF inputs to allow for the smooth double spending of 0-conf transactions.

The upcoming version (0.96) supports CPFP to help you "pull" transactions in. RBF support is planned for a later version.