Can this be used to (quickly) connect to pieces of base molding together?

The Loctite Ultra Liquid Control Super Glue more than likely will not be suitable in this application. The glue requires that the two pieces being joined must fit precisely with no gaps. The success of the bond will also depend upon the composition of the molding and its porosity. Another variable to consider is the amount of expansion/contraction (stress) the final bond will be subjected to. Unfortunately, without more information about the type of molding or project we cannot make a good recommendation at this time. Please contact the Help Line at 1-800-624-7767 so we can better help you.