We present the growth of homogeneous InAs1-xPx nanowires as well as InAs1-xPx heterostructure segments in InAs nanowires with P concentrations varying from 22% to 100%. The incorporation of P has been studied as a function of TBP/TBAs ratio, temperature, and diameter of the wires. The crystal structure of the InAs as well as the InAs1-xPx segments were found to be wurtzite as determined from high-resolution transmission electron microscopy. Furthermore, temperature-dependent electrical transport measurements were performed on individual heterostructured wires to extract the conduction band offset of InAs1-xPx relative to InAs as a function of composition. From these measurements we extract a value of the linear coefficient of the conduction band versus x of 0.6 eV and a nonlinear coefficient, or bowing parameter, of 0.2 eV. Finally, homogeneous InAs 0.8P0.2 nanowires were shown to have a nondegenerate n-type doping and function as field-effect transistors at room temperature.