We're OK (in the theoretical sense) if the relevant QM theory is special-relativistic. The items are prepared in the same state in their individual centre of m

We're OK (in the theoretical sense) if the relevant QM theory is special-relativistic. The items are prepared in the same state in their individual centre of momentum (COM) frames, and in those the (frame-dependent) quantities you are worried about vanish. The COM frames are related by a Lorentz transformation. The COM frame is available thanks to the Lorentz group's boosts. The <https://en.wikipedia.org/wiki/Lorentz_group https://en.wikipedia.org/wiki/Lorentz_group> at every point defines flat spacetime (General Relativity guarantees a small patch of flat spacetime around every point in a general curved spacetime), and relativistic quantum theories (of which the Standard Model is an example) have the Lorentz group fundamentally "baked in". Further detail: <https://en.wikipedia.org/wiki/Center-of-momentum_frame https://en.wikipedia.org/wiki/Center-of-momentum_frame> and in a somewhat scattered usenet thread <https://math.ucr.edu/home/baez/boosts.html https://math.ucr.edu/home/baez/boosts.html>. Experimentally one could set up a careful filtering mechanism (polarizers, traps, and so on). Each such particle will be in the measurement basis with 100% probability. We can attach a COM frame to each, and relate the COM frame and the laboratory frame to one another through a transform.