Courts do make orders for parents to share driving and there is no blame
associated with who moved. In a similar way a man having an affair which caused the divorce isn't asked to pay more.
Contact is for the benefit of children and a judge would look at resources of each parent to determine what might be the best outcome that supports the DC seeing both parents. An example if the nrp couldn't drive for some reason does that mean the DC don't see that parent? If the mum refuses to step up with driving on principle rather than having valid reasons a court would frown on this.
If the RP has time, ability and money to assist with drop offs then it could very well be ordered. Welfare and safety of nrp to drive especially if he drives for work would be looked at as well. Fatigue and driving would be a genuine safeguarding issue.
As someone mentioned the contact every weekend is not usual. Was this agreed via mediation? The mother could argue that she would be prepared to drive one way twice a month rather than 4 times a month and contact could be reduced.
However I would advise against court unless absolutely necessary as it will cause such animosity that is rarely ever recovered from.
How old are the DC? Is public transport an option at some stage? Can your partner make adjustments to his work so that Fridays are work from home or finish early. Has he asked the mum to drive to somewhere along the route which might assist.
I would try these avenues first, formally ask for her assistance in writing, if driving safety is a genuine issue and invite her to mediation to discuss if a compromise could be found.
Is contact court ordered?