Young Couple Paying Off Mortgage

I recently spoke to a past client who is looking to buy a new home and wants to use the equity from their current home to try to pay cash and have no mortgage.  They want to look into a bridge loan or HELOC and then pay those off once the home sells.  The move is more of a lateral move in price point.  We discussed their reasoning behind this desire and they are open to looking at other options, but I can tell they have doubts.  They are really young and I think there are better options for them.  I also think this would be a great case study to share with financial advisors.  I would love to get on a call to discuss and get some help on showing them why investing more of their equity and getting a mortgage could be a better option for them.