Cut the cream cheese into small pieces and place in a food processor or blender. Add the sour cream, half-and-half, sugar, and salt; purée until smooth. Chill the mixture for at least 8 hours in the refrigerator.
When you're ready to make the ice cream, churn it in your ice cream maker according to the manufacturer's instructions. While the ice cream is churning, place the cake crumbs in a large bowl or in the container you are going to use to store the ice cream. Once the ice cream is finished churning, add it to the cake crumbs and quickly fold the red velvet cake crumbs into the ice cream. Place the container in the freezer for at least a few hours to allow the ice cream to harden.