I'm with skipy.

Practice desensitises you. Or better... helps you push the feelings deeper down so they won't show on the surface but can be felt in the music.   

I had this problem with a number of songs I sang to or with my daughter. After she passed away I would choke up. I still do on some. And sometimes that is not a bad thing.

The way I sing Summertime and Motherless Child now has so much more impact than they did when she was alive. There is an underlying current of experience that (I hope) enriches the song coming out of me and is noticed by audience.