Multiple SQL queries fired asynchronously progressively slow down
In order to understand what is happening, I would gather more metrics. First, don't use Profiler, it adds it's own overhead. Assuming you're on 2012 or better use Extended Events. Further, using Extended Events create a session that not only captures rpc_completed and batch_completed for your queries, but also add wait_info_external and wait_completed. With all this add causality tracking. Here you'll be able to see exactly what the query is waiting on as well as how long it takes to run.

On top of this, get the execution plans for when it runs well and when it runs slowly. Compare the two to find any differences.

All this should show you where the problem lies.