Flexible high-density mapping catheter tips and flexible ablation catheter tips with onboard high-density mapping electrodes are disclosed. These tips can be used for diagnosing and treating cardiac arrhythmias. The flexible, distal tips are adapted to conform to tissue and comprise a plurality of microelectrodes mounted to permit relative movement among at least some of the microelectrodes. The flexible tip portions may comprise a flexible framework forming a flexible array of microelectrodes (for example, a planar or cylindrical array) adapted to conform to tissue and constructed at least in part from nonconductive material in some embodiments. The flexible array of microelectrodes may be formed from a plurality of rows of longitudinally-aligned microelectrodes. The flexible array may further comprise, for example, a plurality of electrode-carrying arms or electrode-carrier bands. Multiple flexible frameworks may be present on a single device. A delivery adapter having an internal compression cone is also disclosed.