See Shake The Dust From Your Feet - P1.
Turning back to the prerogatives enumerated in the Synoptic Gospels, we see that they align very closely with the specific ordinances required of those who would go up to the Temple Mount. Setting aside possible exceptions introduced in Mark, we see that both instruct against taking staves, donning any kind of shoe, carrying vestments for the storage of money (and probably other items), and wearing an additional outer garment. Both also require the removal of dust from one’s feet. These prohibitions were probably meant to remind people of the sanctity of the Temple and the service they were performing. This was a sacred enterprise. Just as they should not think to disregard the Temple and use it as a shortcut if they are engaged in common activity, so when they are engaged in sacrosanct activity, they should not bring things with them that might pollute their course. For instance, those who carried a staff, which was often used for journeys, might be tempted to do a bit of extra traveling on their way to the Temple and those who carried bags of money might be tempted to engage in commerce. Removing dust from one’s feet symbolized separation from the unclean and removing one’s sandals or shoes symbolized the willingness to be holy.[5]
The significant difference seems to be one of location. The Rabbis state that one is not to spit on the Temple Mount or use it as a shortcut, which is not incorporated into Yeshua’s directives. However, Yeshua’s disciples won’t be on the Temple Mount to spit there or to use as a shortcut. That could explain why those two were left out of Yeshua’s commands (of course, there’s no reason why Yeshua should include all of these Rabbinic ordnances), but we still don’t know why Yeshua would apply the others to his disciples when they are also not specifically journeying to the Temple Mount. They are going out into the world. Perhaps even away from the Temple Mount. I believe the answer can be found in an understanding of Yeshua’s mission and message.
Yeshua was reforming Israel around himself and was therefore redefining what it meant to be the children of YHWH.[6] In doing so, he subverted established and authoritative systems, traditions, and symbols. That does not mean he was doing away with Judaism for something else or critiquing it from the outside, only that he was using Jewish elements in different and (so he believed) divinely authorized ways. One of those elements was the status and concept of Temple. According to ancient tradition, the Temple was already itself a subversion and redefinition of a former concept: Tabernacle. After the tribes of Israel had settled in the land, but before Solomon’s Temple was constructed, the Hebrew scripture seems to indicate that the house of God
may have been established at Shiloh. It probably would have taken another redefinition or subversion of the way of things in order to relocate it to Jerusalem. If some scholars are to be believed, it was thanks to one or more scribes at the time of King Josiah that Deuteronomy or the core of it was discovered
[7] (i.e., purposely written or assembled) in order to legitimate political and/or religious reforms by the nation’s Sovereign (specifically to eliminate the practice in Israel of any religion other than that of YHWH and to centrally localize that religion in the Jerusalem city and temple).[8] Using religious texts or even finding religious texts in order to support legislative or governmental projects is a practice common both to modern and ancient times.[9] Once the Temple along with hope of its restoration had been totally destroyed, the Rabbis further reformed and redefined things so that a Temple was entirely superfluous. The Shekinah now dwelled in them as it once had within the Temple: If ten men sit together and occupy themselves in Torah, the Shekinah rests among them.
[10] Indeed, they argued the divine presence that used to fill the Temple would come into even a single person and defended it with Exodus 20:24: in every place where I record my name, I will come into you.
[11]
See Shake The Dust From Your Feet - P3.
[5] – We are reminded of YHWH’s command to Moses in Exodus 3:5: Take your sandals off your feet, for the place on which you are standing is holy ground.
[6] – Reforming and redefining Israel
was a very Jewish thing to do. At some point, various Semitic tribes with only a common, distant ancestry came under the same covenant and legislation. Moses is attributed with this major reform. Another great reformer would be the king who took those tribes with their own political sovereignties and united them into a national entity—commonly identified in scripture as David. In Greco-Roman times, we see people like the Essenes, who considered themselves Sons of Light and everyone else either Sons of Darkness or followers of Belial, the Samaritans, who believed themselves to be the true people of God preserving his true commands at Mount Gerizim, and countless others.
[7] – The discovery of the Scroll of Torah
and Josiah’s Reform is found in 2 Kings 22-23.
[8] – See the theory of The Deuteronomistic History. A short book that looks at the ideas and arguments while trying to pave a way forward through the many difficulties is Thomas Romer’s The So-Called Deuteronomistic History. K.L. Noll recently challenged the entire hypothesis in a ground-breaking article in the Journal for the Study of the Old Testament, Deuteronomistic History or Deuteronomic Debate? (A Thought Experiment).
[9] – Many ancient Near Eastern kings claimed to have discovered religious texts in their god’s temple in order to support religious, political, or social actions and reforms that, in reality, were probably not actually inspired by those texts.
[10] – Mishnah, Tractate Aboth 3:6
[11] – בכל המקום אשר אזכיר את שמי אבוא אליך - The Hebrew for you
is singular.

Chase Says:
You’ve got an interesting take on this pericope.
In context, it seems that Mark uses the passage for the same purpose as Matthew and Luke, but the actual content is quite different than the other two synoptics. I would guess that Matthew and Luke change the details of the command in Mark either to draw parallels to temple tradition (as you have pointed them out), or because they think the almost complete lack of provisions makes more sense in the narrative.
In all three narratives (as best I can remember without re-reading Luke right now), Jesus is sending his apostles into Jewish communities to heal and preach. In Mark — who has Jesus telling the apostles to bring a staff and wear sandals, but not to bring money or food — the purpose of the command isn’t explained. Matthew, however, greatly expands Mark, and makes the directive’s reason much clearer; Jesus implies that they’re testing the hospitality of the Jews in the villages they’re sent to.
My sense would be that Matthew explicates Mark’s purpose correctly. The idea that the apostles are testing Jewish receptiveness to Jesus and his gospel of repentance fits in well with Mark’s narrative, in which Jesus is repeatedly rejected by the Jews before finding more acceptance among the gentiles, so the question of why Matthew changes the command itself still remains. I think you’re onto something due to the specifically Jewish context of the pericope in all three synoptics. The author of Luke-Acts provides an interesting tidbit for us in Acts 13:51, where he depicts the apostles shaking the dust off their feet "in protest" of the Jews who incited persecution against them, vs. the gentiles who were receptive to their message.