  Novel Ni-Co-Mn-Ti all-d-metal Heusler alloys are exciting due to large
multicaloric effects combined with enhanced mechanical properties. An optimized
heat treatment for a series of these compounds leads to very sharp phase
transitions in bulk alloys with isothermal entropy changes of up to 38 J
kg$^{-1}$ K$^{-1}$ for a magnetic field change of 2 T. The differences of
as-cast and annealed samples are analyzed by investigating microstructure and
phase transitions in detail by optical microscopy. We identify different grain
structures as well as stoichiometric (in)homogenieties as reasons for
differently sharp martensitic transitions after ideal and non-ideal annealing.
We develop alloy design rules for tuning the magnetostructural phase transition
and evaluate specifically the sensitivity of the transition temperature towards
the externally applied magnetic fields ($\frac{dT_t}{\mu_0dH}$) by analyzing
the different stoichiometries. We then set up a phase diagram including
martensitic transition temperatures and austenite Curie temperatures depending
on the e/a ratio for varying Co and Ti content. The evolution of the Curie
temperature with changing stoichiometry is compared to other Heusler systems.
Density Functional Theory calculations reveal a correlation of T$_C$ with the
stoichiometry as well as with the order state of the austenite. This combined
approach of experiment and theory allows for an efficient development of new
systems towards promising magnetocaloric properties. Direct adiabatic
temperature change measurements show here the largest change of -4 K in a
magnetic field change of 1.93 T for Ni$_{35}$Co$_{15}$Mn$_{37}$Ti$_{13}$.
