  Recently there has been an increasing interest in the study of generalized
symmetries in dimensions higher than two. This has lead to the discovery of
various manifestations of generalized symmetries, notably higher-group and
non-invertible symmetries, in four dimensions. In this paper we shall examine
what happens to this structure when the 4d theory is compactified to lower
dimensions, specifically to 3d and 2d, where we shall be mainly interested in
generalized symmetry structures whose origin can be linked to mixed
flavor-gauge anomalies. We discuss several aspects of the compactification, and
in particular argue that under certain conditions the discussed generalized
symmetry structure may trivialize in the infrared. Nevertheless, we show that
even when this happens the presence of the 4d generalized symmetry structure
may still leave an imprint on the low-energy theory in terms of additional 't
Hooft anomalies or by breaking part of the symmetry. We apply and illustrate
this using known examples of compactifications from four dimensions,
particularly, the reduction of 4d $\mathcal{N}=1$ $U(N_c)$ SQCD on a circle to
3d and on a sphere to 2d.
