The Sunnah is for the muezzin to give the adhaan from a high place, raising his voice, elongating the vowels, facing the qiblah and turning his head to the right and left when saying “Hayya ‘ala al-salaah” and “Hayya ‘ala al-falaah.”
It is not Sunnah for him to walk whilst giving the adhaan, because that is not known in the Sunnah, either from the words or approval of the Prophet (blessings and peace of Allah be upon him). If he does that it is disliked (makrooh), but the adhaan is still valid, because the aim is to convey the call to prayer, and that has been done.
Al-Mardaawi (may Allah have mercy on him) said:
If he gives the adhaan or iqaamah sitting or riding with no excuse, or walking, it is permissible but is makrooh according to the sound view. End quote from al-Insaaf, 2/161. See also al-Majmoo‘ by al-Nawawi, 3/108.
It is more makrooh if he walks a great deal. Ibn Haamid (may Allah have mercy on him) said: If he gives the adhaan sitting or walking a great deal, it invalidates (the adhaan). End quote.
See: Ahkaam al-Adhaan wa’l-Iqaamah, by Saami ibn Farraaj al-Haazimi, 218-219
If he decides to change his place for a valid reason or in the interests of the adhaan, e.g., so that he could be heard from another direction or so that people who cannot hear him may see him, and the like, and he walks a little, there is nothing wrong with that and it is not makrooh in this case.
See also the answer to question no. 142863.
And Allah knows best.