It is a Sunnah to trot in the first three circuits of tawaaf and it is Sunnah to jog in saa‘i between the two markers; these are two Sunnahs that apply to men, not women.
If a man is accompanying a 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 Qudaamah (may Allah have mercy on him) said: The tawaaf and saa‘i of women is all walking. Ibn al-Mundhir said: The scholars are unanimously agreed that women do not have to trot around the Ka‘bah 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 (tawaaf and saa‘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. End quote from al-Mughni, 3/197
Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) was asked: You referred to jogging in tawaaf and saa‘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 tawaaf or saa‘i. I used to think initially that a woman should hasten between the two markers in saa‘i, i.e., jog, because the origin of saa‘i is the story of the mother of Ismaa‘eel [and he related the story of Haajar]. 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 tawaaf we will meet at Maqaam Ibraaheem; 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. End quote from al-Liqa’ al-Shahri, 7/21; Majmoo‘ Fataawa 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 tawaaf riding.
And Allah knows best.