What’s the evidence?
It seems that the notion that physical activity involving limbs “crossing the midline” is of great neurological benefit may not be based on any experimental evidence. It looks like it is based on the observation that delayed development in children can result in young children not using a dominant hand to pick something up “across the midline” but to continue (as young infants do) to pick things up with the nearest hand; Occupational Therapists alegedly therefore feel that development can be fostered by training children with delayed development to pick things up “across the midline”. (This does sound like a blatant case of addressing the symptom rather than the problem!!) So it is possible that all of our assumptions about the usefulness of such exercises (and I remember seeing them in the classroom in my school over 5 years ago) are only (possibly) useful in helping chidlren with delayed development to develop a dominant hand!!!
This is one 2017 review of the literature (which didn’t come up with any research into this effect!)…