Ruling on the muezzin taking a few steps whilst reciting the adhaan

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. 

Al-Mubdi‘, 1/270 

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.