  We address the electronic properties of quantum dots in the two-dimensional
$\alpha-\mathcal{T}_3$ lattice when subjected to a perpendicular magnetic
field. Implementing an infinite mass boundary condition, we first solve the
eigenvalue problem for an isolated quantum dot in the low-energy,
long-wavelength approximation where the system is described by an effective
Dirac-like Hamiltonian that interpolates between the graphene (pseudospin 1/2)
and Dice (pseudospin 1) limits. Results are compared to a full numerical
(finite-mass) tight-binding lattice calculation. In a second step we analyse
charge transport through a contacted $\alpha-\mathcal{T}_3$ quantum dot in a
magnetic field by calculating the local density of states and the conductance
within the kernel polynomial and Landauer-B\"uttiker approaches. Thereby the
influence of a disordered environment is discussed as well.
