I think that I understand what you’re asking… you’re trying to track the propagation of the transaction across the network, and compare it with a plain direct message exchange.
I’m afraid that Algorand doesn’t support that right now, and I don’t expect to have this type of feature in the coming future. The Algorand node would (naturally) try to synchronize its transactions across the network, however, the mechanics of that are not expected to be exported and could change over time.
The Algorand blockchain ( like many other blockchains ) is focused around ensuring that your transaction would make it into a block ( eventually ). The propagation of the said transaction would likely to be optimized around network throughput rather than latency.