Strain engineering in single-crystalline oxide membranes offers a versatile platform to tailor functional properties beyond thin films or bulk materials. However, accurately determining strain transfer and distribution within these membranes remains challenging, limiting direct correlations between structural distortion and functional response and hindering the rational design of flexible devices. Here, high-resolution synchrotron x-ray diffraction is used for in situ monitoring of strain in (001)-oriented La_0.7Sr_0.3MnO₃ membranes on flexible polymer substrates under stretching and bending. This approach enables quantitative determination of in-plane strain as both macroscopic averages and spatially resolved maps. Macroscopic analysis shows that strain transfer is most efficient in thinner films and leads to distinct strain symmetries under stretching and bending. Spatially resolved measurements reveal local strain heterogeneity that increases with applied stress beyond macroscopic averages. Correlating strain distribution with estimated Curie temperature shifts illustrates how such heterogeneity could translate into spatially non-uniform functional behavior. Additionally, the setup's sensitivity to crystallographic orientation enables analysis of stacked and twisted architectures, where interlayer strain transfer is effective yet attenuated with distance from the substrate. Altogether, this framework establishes a robust approach for probing oxide membranes, opening pathways toward strain-engineered flexible devices, multilayer heterostructures, and operando investigations.