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Answer: We have good news and bad news. The bad news is that we were not able to find any classical or semi-classical sources that there is a mitzva for every 4 amot one walks in Israel. The good news is that there are bigger and better ways to get that effect, which those of us who live here do naturally. On to the sources!
Bemare Habazak - Rabbis Questions (654)
Rabbi Daniel Mann
337 - The Nature of the Fulfillment of the Mitzva of Mezuza
338 - Walking in Israel
339 - Continuing to Eat a Seuda Shlishit without Bread
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The Rambam (Melachim 5:11) paraphrases this gemara. However, his language indicates that this source may not be relevant to your question. The Rambam starts with the great merit of living in Eretz Yisrael and then continues: "… even if one walked in it 4 amot, he will merit the World to Come." Thus, someone who lived in chutz la’aretz, took one trip to Israel, landed at Ben Gurion, walked a few steps, and took the next plane out gets this merit. He presents it as (obviously) a (significantly) lower fulfillment of connection to the Land than living there. The question is: if one is living in Israel and meriting extreme spiritual benefits (and has already walked hundreds of thousands of amot), does he get an additional mitzva for walking another four?
HaRav Yehuda Shaviv (Techumin (XXIII, p.)) and HaRav Shlomo Aviner (cited in Shut Eretz Yisrael, 44) assume that a tiyul in Eretz Yisrael is a matter of mitzva and between them cite a few sources: the above gemara/Rambam; a letter by HaRav Tzvi Yehuda Kook; and Mishneh Halachot (III:189). The latter source is dealing with a different question. Is it a mitzva for a ben chutz la’aretz to visit Israel? Some prominent sources posit it is not. The Maharit (II:28) says that if one made a neder to visit Eretz Yisrael, it can be nullified according to the rules for a non-mitzva vow, for there is a mitzva to live in Eretz Yisrael, not to visit it. Also, one is allowed to embark on a voyage by sea on Friday only for a mitzva, and there is a difference of opinions if visiting Eretz Yisrael counts (see Magen Avraham 248:15; Mishna Berura 248:28). The mainstream opinion to reconcile the "non-mitzva" sources with the gemara granting importance to even a "4 amot visit" is that it is not a mitzva per se, but it is nonetheless very worthwhile.
So there is a great spiritual jump when one who had no physical connection came and walked in the Land. But there are two ways to learn the gemara regarding one who already has a great connection by living in Israel (or, to a lesser degree, being in the midst of an extended stay). It is possible that walking more furthers it (1000+1>1000). The other approach is that 0 to 1 is a great jump, but that for one who lives every (or most) breathing moment of his life in Israel (and hopefully contributes to its flourishing), caring about a few more steps is missing the point. (Compare to one who wins a huge lottery and cares about the cents at the end of the multi-digit number.) While one can argue that approach #1 is correct, it is hard to claim that the gemara proves it.
Even according to approach #2, traveling in the Land is significant. Appreciating Eretz Yisrael is important (see Ketubot 111a-b) and may even be connected to the mitzva to live in it (Eretz Hemdah I,1). Seeing sacred, beautiful, … parts of the country promotes appreciation, and the more, the better. This is what Rav Tzvi Yehuda and others refer to. But it should make no difference if this enhanced connection/appreciation came on foot, by car, by going somewhere new, repeating an old visit, or thanking Hashem for Israel when you go to bed. The sources do not seem to indicate that walking per se is a mitzva.
Lessons
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Kuzari -Rabbi Ari Shvat Kuzari class 9 - "Seeing is Believing" (parag. 21-30)
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