This relative is in the 90s and there have been several incidents where there have been falls. The last one resulted in the person being in the nursing home for 3 months. Do not want to deny access to relative, just transporting in any manner in Texas.
If we can just get the picture straight: you have power of attorney or someone else has? For your grandmother? And who is in his/her nineties, the person you have POA for or the relative who wants to take her out? - or both, indeed?
But the issue is: if the elderly person, while elderly, nonetheless has all her mental faculties about her, then it is up to her where she goes and with whom. End of story. You can advise, you can strongly recommend, you can express yourself as forcefully as you like to both the elderly person and the adventurous visitor - you can even menace the visitor that you will hold him personally responsible for any accident that befalls your loved one (this has no legal basis but it might take moral effect) - but you cannot stop her going out if she wants to.
If, however, the elder loved one has dementia and is deemed not to have legal capacity, then it's a different issue. It is your responsibility, as her POA, to act in her best interest and refuse to allow her to go out unless accompanied by an appropriate, responsible caregiver. You should still take her wishes into account, but by welcoming the visitor into her home you're half-doing that anyway, and in any case if her wishes endanger her then you would be obliged to overrule them.
I know whomever is visiting is thinking they are doing a good deed taking the elder out for a change of scenery are not fully aware of what medical mishaps can happen, like falls.
Does this elder live alone at home? Are there any caregivers helping? If the elder is falling when out and about, he/she is probably also falling at home.
The only time I let my own Dad get away from his house was when his paid Caregiver drove Dad to where ever he wanted to go. The Caregiver knew the proper way of getting Dad out of the vehicle, and made sure he was with his walker at all times. There was never a fall.