Sorry for the late response.
NB: While I'm not so sure about the uniformity of the thing assembly procedure across period, time, area, and further, hierarchy of local assemblies in Viking Age and Medieval Scandinavia as some recent scholarship allegedly presuppose, at least on based on fragmentary nature of often very later sources, I'll attempt to focus on some very basic features on "thing assembly site as a space of peace (griðr or friðr)" element below.
If you are familiar with this topic, please skip the following the chapter 1 and directly to its 2 for reading.
1: Physical and symbolic landscape of the thing assembly site
An erected rune stone of the early 11th century central Sweden states that:
"Ulfkell(?) and Arnkell and Gýi, they made the Assembly-place (þikstaþ) here … No landmarks will be more (great), than (the ones) the sons of Ulfr made in (his) memory; able lads in memory of their father (U225, dated around a generation around 1000 CE: the translation as well as the dating is just taken from Database Runor)."
Another famous rune stone (or one of very famous group (ca. 20) of rune stones, called Jarlabanke Runestones - now even with their own wikipedia entry in English as well as tripadvisor page!), presumably erected roughly a generation after, also confirms that elites involved with the set up of the assembly site, or its landmark(s):
"§A Jarlabanki had this stone raised in memory of himself while alive. He alone owned all of Tábýr. May God help his spirit. §B Jarlabanki had this stone raised in memory of himself while alive, and made this Assembly-place (þingstað þenna gærði), and alone owned all of this Hundred (U112)."
You might also notice that Chieftain (local magnate) Jarlabanki also explicitly declares in the inscription that he is a Christian in the early to the middle of the 11th century Central Sweden a bit north to now Stockholm. Sorry for digress.
In addition to runic stones, other landmarks like a mound (note that it was not always - or rather only in rare cases the burial mound) sometimes signified the landscape of the assembly site. Sanmark argues that they served for competing local elite family as a means of theatrical display as well as the commemoration monument of the authority in the local community (Sanmark 2015: 96f.).
It was not only this kind of core landmarks that attached to the assembly site, however. Sanmark also suggests that the site might often be enclosed with stone settings (you can see a possible example in the picture in the first linked assembly site for U 225), wooden palisade, of streams (naturally running ones like a river might have been preferred), citing the possible allusion in Eddic Poetry Grimnir's Saying (Grímnismál), St. 29 (though recorded in much later period):
"Kormt and Ormt and the two Kerlaugar, these Thor must wade every day, when he goes to give judgement [at the thing-assembly site] at Yggdrasill's ash, for the Aesir's bridge burns all with flames, the sacred waters boil (Larrington trans. 2015: 52)."
2: Assembly site as a enclosed place that the peace prevailed
A few recent scholarship on the interdisciplinary assembly site study like Sanmark and Sundqvist emphasize the ritual at the site as a theatrical display of local chieftain's authority. In this context, Thor's movement into the enclosed assembly site (crossing the boiling the sacred water marking the space for peace) cited above could indeed be a rite of passage in a literal sense into the assembly site at the beginning of the assembly (Sanmark 2017: 100, 103).
The next stanza of Grimnir's Saying (St. 30) also continues to narrates on the riding arrival of the Aesir to the assembly site:
"Glad and Golden, Glassy and Skeldbrimir, Silvertuft and Sinir, Gild and Hidden-hoof, Goldtuft and Lightfoot, these horses the Aesir ride every day, when they go to give judgements at Yggdrrasill's ash (Larrington trans. 2014: 53)."
Sanmark also notes that both Eddic poetry and later law texts emphasize the motion of "sitting" in the assembly as a prerequisite for the person's function of the judge, in contrast to the standing speaker who addressed at the assembly (Sanmark 2017: 109f.). According to her, the stone setting at Arnkel's assembly place (U 225) could indeed be a base of wooden bench for judges foundation rather than an enclosure itself.
Thus, the arrival and the sitting of the local elites as judges probably marked the beginning ceremony of the assembly in many cases.
Icelandic Lawbook Grágás indeed has a stipulation on the arrival time for chieftains stricter than other attendees:
"All chieftains are to come to the Assembly before the sun leaves Þingvöllir on the Thursday when ten weeks summer have passed, and if they do not come in this way they pay fines and forfeit their chieftaincy unless some necessity occurs to prevent their coming (Dennis et al. trans. 1980: 57)."
On the other hand, Frostathing Law from late 12th century Northern Norway is known to make an addition of ringing the church bell, very audible element, to the opening ceremony:
"It is the counsel and the command of Archbishop Eystein (d. 1188), to which the most discreet men have agreed, that all shall go fasting to the thing and shall appear at the thing when the sun is due east (0600 in the morning) and shall remain at the thing till noon. And the priest whose duty is to interpret the law book shall have the great bell rung when he is about to go to the thingstead with the book; and while the thing remains in session that bell shall be rung for no other purpose (Frostating Law I-3, the translation is taken from: [Larson trans. 1935: 223f.])."
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