Colloidal quantum dots (CQDs) attract worldwide scientific and technological attention due to the ability to engineer their optical properties by the variation of their size. However, several important applications, such as biological tagging and photovoltaic cells, impose a limit on their size yet demand tunability and thermal stability of the optical band edge. This work introduces a new class of heterostructures, composed of PbSe or PbSe(y)S(1-y) cores, coated by PbS or PbSe(x)S(1-x) shells, with different core-radius/shell-width division, with a radial gradient composition (with 0 < y < 1, 0 < x < 1), which offer a control of the band edge properties by varying the CQDs' composition. Continuous-wave and transient photoluminescence measurements over a wide temperature range (1.4-300 K) revealed a distinct behavior of the heterostructures with respect to that of pure PbSe cores: (i) increase of the emission quantum yield; (ii) red-shift of the absorption edge but a decrease of the emission Stokes shift; (iii) alleviation of a dark exciton recombination, viz., a reduction of an exchange interaction; (iv) tuning of the radiative lifetime with shell width and composition; (v) reduction of the band edge temperature coefficient, dE/dT, viz., induction of thermal stability. The k·p envelope function calculation, considering abrupt or smooth alloying continuation of the potential at the core-shell interface, revealed a delocalization of the hole wave function over the entire volume of the CQDs, as a partial explanation for the marked tunability, nonetheless preserving a desired size.