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Understanding and controlling the director field configuration, shape, and orientation in nematic and cholesteric liquid crystals is of fundamental importance in several branches of science. Liquid crystalline droplets, also known as tactoids, which spontaneously form by nucleation and growth within the biphasic region of the phase diagram where isotropic and nematic phases coexist, challenge our current understanding of liquid crystals under confinement, due to the influence of anisotropic surface boundaries at vanishingly small interfacial tension and are mostly studied under quiescent, quasi-equilibrium conditions. Here, we show that different classes of amyloid fibril nematic and cholesteric tactoids undergo out-of-equilibrium order-order transitions by flow-induced deformations of their shape. The tactoids align under extensional flow and undergo extreme deformation into highly elongated oblate shapes, allowing the cholesteric pitch to decrease as an inverse power law of the tactoids aspect ratio. Energy functional theory and experimental measurements are combined to rationalize the critical elongation ratio above which the director-field configuration of tactoids transforms from bipolar and uniaxial cholesteric to homogenous and to debate on the thermodynamic nature of these transitions. Our findings suggest new opportunities in designing self-assembled liquid crystalline materials where structural and dynamical properties may be tuned by non-equilibrium phase transitions.