Some Quot schemes in tilted hearts and moduli spaces of stable pairs

For a smooth projective variety $X$, we study analogs of Quot functors in hearts of non-standard $t$-structures of $D^b(\mathrm{C}\mathrm{o}\mathrm{h}(X))$. The technical framework is that of families of $t$-structures, as studied in arXiv:1902.08184. We provide several examples and suggest possible directions of further investigation, as we reinterpret moduli spaces of stable pairs, in the sense of Thaddeus (arXiv:alg-geom/9210007) and Huybrechts-Lehn (arXiv:alg-geom/9211001), as instances of Quot schemes.