the exercise of visitation
exercising visitation

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Note added at 14 hrs (2022-12-21 11:14:15 GMT)
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VISITATION TRAVEL COSTS

...The appellate court affirmed the reduction in his child support payments from $1500/mo. to $1087/mo. and also reduced the percentage that he had to pay for the cost of international transportation involved in the exercise of visitation. The court stated that cost-sharing for travel is a child support issue, not a custody issue...
https://www.nclamp.gov/publications/silent-partners/visitati...

By considering separately the jurisdiction requirements for custody determinations (which by definition included visitation rights) and allocation of a parent's travel expenses to exercise visitation rights, the supreme court indicated that the allocation of a parent's travel expenses for exercise of visitation rights was not part of a custody determination under the UCCJA.
https://casetext.com/case/in-re-of-380?__cf_chl_tk=II10By1Ok...

...the parent exercising visitation pays for the transportation costs.
https://legalhearsay.com/who-pays-for-visitation-expenses-wh...

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Note added at 14 hrs (2022-12-21 11:26:54 GMT)
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Według Hague Convention
the exercise of contact (rights)
https://assets.hcch.net/docs/aabaa1c1-4fb5-4c24-92bd-75a8b16...

UK
For example if a parent is exercising contact on an alternate weekend basis with collection of the child from school and returned to school on a Monday, it would not be unreasonable to assume that that parent would undertake the same actions as the other would to ensure that the child’s school uniform is properly cleaned and available to the child to return to school on the Monday.
https://www.stephens-scown.co.uk/family/children-issues/chil...

Shared parenting is taken to include parents exercising contact with their children. 
https://www.drummondmiller.co.uk/news/2020/12/contact-arrang...

https://www.lawscot.org.uk/members/journal/issues/vol-47-iss...

47. District Judge Bowman held that the mother's plan to relocate to the Isle of Wight was reasonable and realistic. She considered the mother's budget of just under £2,500 a month; it could be reduced in most areas. The father offered £200 per month. She considered what she described as the father's well prepared binder of late-lodged documents. She was asked to bear in mind the heavy costs incurred in exercising contact.
https://www.familylawweek.co.uk/site.aspx?i=ed1407