A number of the questions discussed in the background of Bounds on the dimension of lineal extensions also make sense to pose for more general k-planes. In particular, it is reasonable to study whether the packing dimension of a union of large subsets of k-planes can increase if each subset is extended to the entire k-plane.
Furthermore, in recent years, there has been an increase in the study of Furstenberg-type sets in higher dimensions. In two dimensions, the (resolved) Furstenberg set problem asked for sharp dimension bounds on sets F with the following property: that there exists a Hausdorff dimension t set of lines each intersecting F in a set of Hausdorff dimension s. In higher dimensions, it is reasonable to generalize "lines" to k-planes, and much less is known in this setting. Some important early bounds are due to Héra-Keleti-Máthé and Héra, and more recent work of Bright-Dhar combines a number of approaches and gives a very good introduction to the problem.