I understand that a block is generated whenever a proof of work is
submitted by a miner; this happens when the miner processes a
That's not correct. The miners doesn't submit a found block (proof-of-work) when he processes a transaction. He does when he found a block hash that was valid.
That can take a long time and in the time he looks for a good block, he will receive probably thousands of transactions.
He will verify those as they come in and put them in the block he is currently working on, all of them, one by one (until the limit is reached). When he then finds a valid block hash, he will submit the entire block consisting of all, or most of, the transactions he received.
So, to answer your questions, transactions do not necessarily get slower over time, on the condition that the upper limit of transactions in a block is high enough. If the limit is too low, there can arise a bottleneck for transactions trying to be verified.
FYI: You can see that all miners have all those transactions in their working block, but only one of them can submit the block. To prevent transactions from occurring twice, once a miner receives a new valid block from a fellow miner, he checks what transactions in his pool are also in that block. He will remove them from his pool and continues mining for the next block with the transactions from his pool that were not in the block of the fellow miner.