There is little help for discerning the Chronicler's understanding of the phrase highlighted by OP. But, there are some observations to draw on:
The Text
The key verse cited has a parallel in Samuel: 1 Chr 17:5 ≈ 2 Sam 7:6. The main differences come at the end of the verse -
1 Chr 17:5b
וָֽאֶהְיֶ֛ה מֵאֹ֥הֶל אֶל־אֹ֖הֶל וּמִמִּשְׁכָּֽן׃
= wāʾehyeh mēʾōhel ʾel-ʾōhel ûmimmiškān
And I was from tent to tent and from dwelling(place).
2 Sam 7:6b
וָאֶֽהְיֶה֙ מִתְהַלֵּ֔ךְ בְּאֹ֖הֶל וּבְמִשְׁכָּֽן׃
= wāʾehyeh mithallēk bĕʾōhel ûbĕmiškān
And I was going about in a tent and in a dwelling(place).
It is widely accepted that the Chronicler is adapting, revising, recasting the work of the historian of 2 Samuel. It is difficult to know what prompted the differences (noted above in bold): the change of preposition from "in" (be-) to "from" (mi-) could be mechanical: see other examples in Friedrich Delitzsch, Die Lese- und Schreibfehler im Alten Testament, nebst den dem Schrifttexte einverleibten Randnoten klassifiziert (Walter de Gruyter, 1920), p. 113. That is, this could simply be occasioned by a scribal slip between Samuel and Chronicles.
The two texts are also different in length: while the Samuel text makes it explicit that the divine presence moved about (mithallēk), the Chronicles version omits that word, but adds "(tent) to tent", likewise (but differently) implying divine movement.
Observing this textual situation suggests that the two versions are articulating roughly the same idea: that God is moving about in a portable shrine at the time spoken of.
The Commentators
Which probably explains why the (slightly odd form of the) Chronicles text attracts very little attention from the commentators. The sense of the Chronicles text was caught quite early by Rashi (12th C.), who wrote:
but I have [gone] from tent to tent and from tabernacle: i.e., and I have walked from tent to tent and from tabernacle to tabernacle: from Gilgal to Shiloh, and from Shiloh to Nob, and from Nob to Gibeon, and although I was wandering from tent to tent and from tabernacle to tabernacle…
The point, then, is not that there were several "tents" or "tabernacles" that God swapped between, but rather the "tent to tent" phrase refers rather to the staging points of the portable shrine: it is made and re-made with every geographical move.
This sort of understanding is also reflected in a brief comment by one of the major Chronicles scholars of recent years, Sara Japhet, I & II Chronicles: A Commentary (Westminster John Knox Press, 1993), p. 330:
II Sam. 7.5 emphasizes that God's dwelling is a tent, not a house, while the emphasis in Chronicles is on the mobile manner of God's abiding with his people; since the people have not yet settled down permanently, God accompanies them in their wanderings.
Summary
In spite of superficial appearances, then, the Chronicles formulation is not about a multiplicity of tents or tabernacles, but of changing places where the one "tent" was to be found.
