Do markerlights put on a unit embarked in a dedicated transport remain on the passengers after the vehicle is destroyed?
No. The markerlights are on the transport, not the unit inside. Remember they are separate units, and generally you can not target units inside a transport (template vs. open topped being the exception). Any markerlights on a destroyed unit are removed with the unit.

The only way that would be possible, is if the targeted unit had a way to embark during your shooting phase. To my knowledge that's not possible.

Note: In 8th edition, this is still true, except that there are no templates anymore.