Both men were politicians first and foremost, and I believe if both were alive today, they would be running around with gay pride flags, defending infanticide without limits, demanding an end to fossil fuels, and being in lock step with every other contemporary anti-American leftist.
I believe this is true based on the behavior of their surviving family members and proteges.
Look no further than Ted Kennedy, Jessie Jackson, and Rafael Warnock for examples.
JFK and MLK shared one trait which led both to their status as icons and to their shared experience of being assassinated. That trait was the ability to receive the support of all the factions in their "party."
JFK had the support of every branch of Democrats at the time, from the Reaganesque faction (which eventually left the party) through the unionists like HHH and the Southern racists like LBJ, all the way over to the McCarthy/McGovern social-libs. After JFK's death, LBJ became President, but the main battle for supremacy in the party was between the HHH unionists and the social-libs: the libs would have won in '68 if RFK hadn't been assassinated, but they won the party in '72 with McGovern, and (not counting a short stint with Carter) have been running the party ever since.
By the same token, MLK was the only black leader in American history to be able to join under his leadership the Booker-T-Washington black conservatives, the WEB-DuBois black social-libs, and the Black-Muslim-Panther-Us-UHURU revolutionaries. After MLK's death, the black conservatives lost all standing in the civil rights movement, which became a vehicle for the social-libs, who have used the violence of the revolutionaries when they have found it useful, and have convinced the non-black population that they and they alone represent the total history of the civil rights movement--including wearing the mantle of MLK. The Bookerite conservatives, like the Reagan Democrats, ended up in the GOP, where they still don't get the credit they deserve.