Transition on the Chorus
Now will get to the good part how to transition the Hip Hop way. This first transition is pretty standard and pretty self-explanatory. You beatmatch your songs and drop your incoming song, which usually would be the intro, on the chorus of the outgoing song, "the live track". You can fade in the incoming song and fade out the outgoing song OR you can just you can just place the fader in middle and wait to until the chorus is over and slide the fader to incoming song which should be right where your 1st verse or 1st chorus comes in.
Transition Squash Technique
Another variation of this technique would be using what I've heard been called a "squash". A squash where you hit the stop button on turntable thereby creating a slight delay effect. You can use the technique same Transition on Chorus but before sliding over the fader to new track . You first hit the stop button on outgoing track right last snare and move the fader over that track and then quickly over to new track.
Transition Backspin Technique
Yet another variation would instead of a squash you use a backspin. A backspin like the name suggests is just spinning the record backwards quickly. From the sample below you should hear to perform this technique.
Transition in Verses
Here is technique not much used and really it can't be used on most songs but it's nice technique to have in your arsenal. Instead of transition on the chorus of the outgoing song, "Live track" you stat your transition toward the end of a verse of the outgoing song.
To make this work you can fade in the incoming track so that by the fourth snare before cutting off the outgoing song you're incoming song is fully heard. Also if you dig deep enough you're find a track where there's no need to fade in. You can just drop the incoming track right on top of outgoing song and it will sound great. You can use the EQ on outgoing track by turning off the lows and lowering the mids to make the live track sound more like acapella to give your incoming track more emphasis.