If this were me, I would try to peel off any caulk that is loose. Use ONLY a plastic tool on the roof membrane and do NOT poke any holes in the roof trying to get at caulk that is not hanging loose.
I would then gently scrub with a soft bristle brush the roof with SPic and Span liquid cleaner and rinse off throughly.
I would then get wither 2", 3" or 4" wide Eternabond tape (from Amazon) after doing some measuring and estimating what you might need. I have used Eternabond tape as a preventive measure on all seems, including the roof edges (front, back and sides) where the roof ends and meets the trim, or side gutters or any other vent.
There are numerous youtube videos on how to apply Eternabond tape.
Use a small amount of Mineral Spirits on a cloth and rub off any dirt/deposits on the roof where you will be applying tape. DO NOT use in excess as the roof will stretch. IMMEDIATELY wipe with generous amount of rubbing alcohol on another cloth and let dry for a few minutes. The Mineral Spirits will take up dirt that cannot be touched by any other cleaner, but must be removed quickly with the rubbing alcohol to prevent roof damage. Do small sections at a time.
Apply Eternabond over seams and old caulk. Wipe tape outer layer with rubbing alcohol and overlap 1/2" if applying a second strip.
This Eternabond should last 10 years or more. Clean and inspect your roof at least every 6 months.
I hope this helps
