This pertains to Insights and is in regard to Rav Moshe OCII:57. "...who follows the lenient ruling of the Rema, adds that even if one is eating fruit (or other items which normally require a new Berachah if one changes places), he does not need to recite a new Berachah in the new location if he continues eating as he goes to the new location and does not stop for a period of k'dei dibur."
First, are we allowed to eat as we walk (not that I don't do myself on rare occasions. Second, what is the principle involved in continuing to walk? Once the person gets to the new location, as quoted above, he does not have to recite another Beracha, so why so even if he stops on the way, since each time he stops he has established a new location?
In reply to your fist question there is no Issur of eating while walking.
(The Shulchan Aruch CM 34:18 rules that someone who eats in public is
Pasul l'Edus because he is a despicable person. This is only true if eating
in public is understood to be improper and an indication that the person has no sense of shame. In our society this would hardly be the case and a person who eats in public would not be disqualified from giving Edus.)
Your second question is the result of unclear wording in our text. It should
read "...and does not stop eating for a period of Toch K'dei Dibur."
Once he stops eating it is no longer considered a continuation and he must
say a new Berachah.
Kol Tuv
Ilan Segal