Will nodes configured with a larger-than-default mempool automatically retransmit transactions that have been dropped from the default mempool? If a transaction is sent and dropped from the default mempool before it is included in a block, but nodes with a larger-than-default mempool size exist that still have the transaction, will those nodes retransmit that transaction eventually?
This is a matter of policy, so different Bitcoin implementations may have different behaviors. I'm most curious about Bitcoin Core, since it's very common, but if other node implementations differ, I'm curious about them too.
If such transactions aren't retransmitted, I'm also curious why not, since it seems like such behavior might be useful.
So far, Bitcoin Core rebroadcasting is a wallet-level responsibility. Transactions that aren't relevant to any of the Bitcoin Core's internal wallet are not rebroadcast at all.
There is work to change that: https://github.com/bitcoin/bitcoin/pull/21061. This makes sense, as right now, observing a rebroadcast is a very clear sign that the transaction belongs to the wallet, which is a privacy leak.
So I think in short the answer is: yes, it makes sense, but so far it hasn't been done.