We propose and investigate a new type of photonic crystal (PhC) cavity for integrated quantum photonics, which provides tailored optical modes with both confined and extended spatial components. The structures consist of elongated PhC cavities in which the effective index of refraction is varied quasi-linearly along their axis, implemented by systematic lateral shifts of the PhC holes. The confined modes have approximately Airy-function envelopes, exhibiting single peaks and extended tails, which is useful for optimizing single photon extraction and transmission in integrated quantum photonic devices. The measured spectrally resolved near-field patterns of such devices show the expected spatial and resonance wavelength behavior, in agreement with numerical simulations of the Airy-Bloch modes. The effects of fabrication-induced disorder on the mode features are also analyzed and discussed. Selective excitation of specific Airy-Bloch modes using integrated, site-controlled quantum dots as localized light sources is demonstrated. Based on the tilted-potential cavity, multiple-QD single photon emitters exploiting wavelength division multiplexing are proposed. (C) 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement