Does a full node attempt to remove tx from a valid block that shares the same parent?

Does a full node attempt to remove tx from a valid block that shares the same parent?

If a full node receives A-B-C with C being a valid block and removes the transactions from the mempool, what happens if the next block is A-B-D? It is my understanding that the full node validates the D block and then if it’s valid hangs on to it until the blocks are reconciled? My question is does the full node try to remove the tx in D from the mempool as well? Since it is likely that C and D share some of the same txs what happens when the node attempts to delete a transaction in D, that was in C from the mempool?

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