In the Name of Allah, Most Gracious, Most Merciful.
All praise and thanks are due to Allah, and peace and blessings be upon His Messenger.
In this fatwa:
If a man is accompanying a woman or an elderly person, and he is afraid he will lose them if jogs in tawaf and sa`i, then he may walk with them and forego trotting and jogging.
Answering your question, the Fatwa Center at Islam Q and A, states:
If a man is accompanying woman or an elderly person, and he is afraid he will lose them if he goes on ahead of them, then he may walk with them and forego trotting and jogging.
Ibn Qudamah (may Allah have mercy on him) said:
“The tawaf and sa`i of women is all walking. Ibn al-Mundhir said: The scholars are unanimously agreed that women do not have to trot around the Kabah or between Al-Safa and Al-Marwah, and they did not have to bare the right shoulder. That is because the original purpose in both (tawaf and sa`i) was to demonstrate strength, and that is not intended in the case of women; and because women should be covered, and trotting and baring the shoulder imply uncovering. (Al-Mughni, 3/197)
Shaykh Ibn Uthaymeen (may Allah have mercy on him) was asked: You referred to jogging in tawaf and sa`i… is jogging only for men? And if a man has a woman or women with him, should they run with him or not?
Some scholars stated that the Muslim scholars are unanimously agreed that women should not jog either in tawaf or sa`i. I used to think initially that a woman should hasten between the two markers in sa`i, i.e., jog, because the origin of sa`i is the story of the mother of Isma`il [and he related the story of Hajar].
But when I saw some of the scholars narrating that there was scholarly consensus that women should walk and not hasten, I saw that the correct view is that women should walk and not run.
There remains the issue of the mahram who is with her: should he hasten and leave her behind, or should he walk with her as she is walking? We say: If the woman can find her way by herself and she has experience and there is no fear for her, there is nothing wrong with the man trotting in the first three circuits and saying to her: After tawaf we will meet at Maqam Ibrahim; but if she cannot find her way by herself and there is fear for her, then it is better for him to walk with her than to jog or walk quickly between the two markers. (Al-Liqa’ al-Shahri, 7/21; Majmu` Fatawa Ibn Uthaymeen, 22/430)
If a sick or elderly person cannot walk normally, there is nothing wrong with him walking slowly, according to his ability. If it is too hard for him to do that, there is nothing wrong with him doing tawaf riding.
Almighty Allah knows best.