a and Katrina rock IPL 2
What does hip-hop sensation Akon do when he gets a moment with the dancing Katrina Kaif? Well, he prefers to sway to her tunes. Both the American singer Akon and Bollywood actress Katrina Kaif were the show-stoppers at the IPL 2 Grand Finale held in Johannesburg, South Africa. Katrina Kaif was seen dancing to the English version of 'Jai Ho' of 'Slumdog Millionaire' fame, the version originally sung by Pussycat Dolls. Akon on the other hand, sang his hit number 'Smack That' in fusion with the 'Raghupati Raghav Raja Ram' bhajan and African drums. While most rocked to the performances, the crowd booed Akon's initial lip-sync.

Could you please explain to me the emboldened parts?

I know "sway" means "to persuade someone", but I wonder what "sway to her tunes" means. Does that mean "He is persuaded to dance to her tunes" Or "He is persuading her to dance to his songs/tunes"?

Also, I would like to know what "lip-sync" means.
I agree that "sway" here means to move with the music. But the implication escapes me.
To "get a moment with the dancing K.K." should mean that he had an opportunity to perform some kind of duet with her. It'sounds like he's a rapper and she's a dancer. But instead of rapping to her dancing, he just swayed with it. Makes no sense.
But I believe "to sway" can mean to yield, or bend, in the intransitive, so perhaps the comentator means he let her carry the show.
I think it means that he "followed her lead" and also acted as a show-stopper, but I do believe it is also a bit of a pun in the literal sense of sway meaning to swing (i.e., dance to her kind of music). I don't know these people, so it's a little difficult for me to get the nuances.

lip-sync means that he wasn't really singing, just moving his lips in-sync to a pre-recorded track as if he was.
 Avangi's reply was promoted to an answer.