Do you have the right trim for your replacement?
I have done lots of this stuff and the trim is usually a nice tight fit and requires no caulking or anything else to hold it in place, including going around corners.
There are usually 2 little lips that stick inside the aluminum channel to hold it in place and you should have to apply pressure to the molding to get it in. That is you should have to bend the center of the trim, or arch it up in the middle to get the second lip into the groove.
Try taking a small one inch piece and sticking it in there. Then wiggle it from side to side. If there is any play in it it could be that the trim is too small.
The heat gun on the corners is a good idea to help it along.
2007 GMC 3500 dually ext. cab 4X4 LBZ Dmax/Allison - 2007 Pacific Coachworks Tango 306RLSS
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