Streaming speeds are a factors of 2 things Size and time. Local connections should have the benifits of both. On a wired Lan connection you could get 100mb/s and less than 10ms latency. That same connection over the internet is reduced to your upload speed and internet latency. So say you have 5mb/s up and 100ms latency remotely. think of latency as how many tries per second to reach the max size. 10ms is .01 seconds 100ms is .1 seconds. When you get 6000k locally its because of 100mb/s possible 6000k times your latency happened in 1 second. Remotely 5mb/s already limits your speed to 5000k if you had the same latency as local. Most of the time your latency is much higher on remote. other things factor as well but I am sure you understand these basics.
wifi locally can reduce the stream speed even at N rates below your internet rate. Wireless g speeds can be 54mb/s and n can be 300mb/s. If your laptop is wireless n and your sling is connected at 100mb/s you can't sling 300mb/s but at wireless n you have increased latency. Ping your sling from the wired lan then wireless lan g and again wireless n. All things equal you will see a average latency much higher in wireless n.