There is a known situation that when you attempt to edit a broadcast in Adobe Premiere Pro you may encounter the video and audio being out of sync. This happens due to the broadcast dropping frames during internet fluctuations and Adobe Premiere Pro not capable of handling variable Frame Rates.
Note: The video editing application Final Cut Pro X does not suffer from this issue. If you have this program available to you that may be a better option.
To fix this issue you'll need to convert the file using a program called Hand Break. It can be downloaded HERE for free.
When you open the software you'll be prompted to choose/open the video file.
Next choose the destination to save the file to. By default it'll save to your Movies folder.
You will now go to the Video tab in the center of the application and choose Constant Framerate. (See Image Below)
Handbrake will begin to convert the video into a new file. Depending on the length of the clip conversion can take a long time. Expect the conversion to last up to the duration of the clip.