  We consider the antiferromagnetic spin-1/2 $XXZ$ Heisenberg model on a
frustrated diamond-chain lattice in a $z$- or $x$-aligned external magnetic
field. We use the strong-coupling approach to elaborate an effective
description in the low-temperature strong-field regime. The obtained effective
models are spin-1/2 $XY$ chains which are exactly solvable through the
Jordan-Wigner fermionization. We perform exact-diagonalization studies of the
magnetization curves to test the quality of the effective description. The
results may have relevance for the description of the azurite spin-chain
compound.
