All the descriptions that I remember of the Logrus describe it as "constantly moving" or similar, does this mean that someone walking it has to move fast before the area they're standing on moves out from under them and, given what happens if you step off the Pattern having started, they get destroyed?
As far as I know, there is no description of someone going through the Logrus in the series (original pentalogies plus Betancourt's trilogy), so we can only speculate.
To elaborate on Zeiss Ikon's answer, I have found occurrences of negotiating, taking and walking the Logrus; all of these terms also apply to the Pattern, so differences between the two probably lie elsewhere.
“At some point, the Pattern discovered—or perhaps possessed all along—the ability to create ‘ghosts,’ short-lived simulacra of those who had negotiated it.”
Prince of Chaos, chapter 4
One of the most distinctive traits of the Logrus is indeed its unstable moving/waving form. It is sometimes described as a maze, which I think hints at how it is traversed:
“They have a sort of equivalent called the Logrus. It’s a kind of chaotic maze. Keeps shifting about. Very dangerous. Unbalances you mentally, too, for a time. No fun.”
Trumps of Doom, chapter 6
“No. The Logrus is—” he hesitated, as if searching for the words to describe the indescribable. “It is a key to finding your way amongst all the Shadow worlds. It is like a maze. By traversing its length, from start to finish, someone born of Chaos may have the Logrus imprinted on his mind forever.”
The Dawn of Amber, chapter 8
Although its most common form is definitely the tentacular shape it takes when it is used to retrieve objects from Shadow, or when summoned.
I closed my eyes and visualized an image of the Logrus shifting, ever shifting. I framed my desire and two of the swimming lines within the eidolon increased in brightness and thickness. I moved my arms, slowly, imitating their undulations, their jerkings. Finally, the lines and my arms seemed to be one, and I opened my hands and extended the lines outward, outward through Shadow.
Trumps of Doom, chapter 6
This thread-like form seems to be close to the nature of the Logrus, as it manifests itself in this way when summoned in Castle Amber, one of the scenes where it's the most tangible. Later on, when cornered by Ghost, the Logrus again forms a tentacle:
Then, as she neared the stairway, the Sign of the Logrus appeared before her, larger than any I’d ever summoned, filling the corridor from wall to wall, roiling, sprawling, fire-shot, tentacular, a reddish haze of menace drifting about it.
Knight of Shadows, chapter 10
“I can offer you cosmic greatness.”
“You already did. I turned you down then, too. Remember?”
“I remember. And I will remember.” A jagged tentacle of the constantly shifting figure moved toward one of the circles of light. There was a blinding rush of flame when they met.
Prince of Chaos, chapter 12
My personal opinion about negotiating the Logrus is that it involves traversing the maze made of these waving threads — exploring the paths to become one with the Logrus' power. Just like the Pattern, but the threads would be made of power, without a physical representation.
Under this assumption, someone walking the Logrus won't have to care about the Logrus moving under them. The maze would just be a metaphor. Although if they lose track of the ever-moving threads, I reckon they'd be destroyed as well. Failing to get through the Logrus is clearly fatal:
“And once in the Courts, you must survive the Logrus. Not all of us do, you know. My brother died on his first attempt. It destroyed him, mind and body. It is not so simple a matter after all.”
The Dawn of Amber, chapter 8
In fact, as an incarnation of Chaos, I see no reason why the Logrus would need a physical representation such as a pattern on the floor of a cavern. From what I recall from the books nothing hints at that.
As all speculations, this theory has its limits. Although I can't picture it, the Logrus is not only tentacular, it also happens to be angular, which makes representations somewhat complicated. :)
“Not really. The Logrus actually alters itself somewhat, constantly. Still, it’s more angular, whereas this is mostly curves and bends.”
Blood of Amber, chapter 9
As I recall from Merlin's descriptions in the second pentalogy, one doesn't so much "walk" the Logrus (as one would the Pattern, as if walking a maze); one "negotiates" the Logrus.
The "original pattern" inside the Jewel of Judgment is described by Corwin as "like a three-dimensional version of the Pattern" and, we later learn, served as the original when Dworkin inscribed the primal Pattern on proto-Kolvir -- in a similar way, the Logrus might be seen as the original from which the proto-Pattern in the Jewel is derived. It might well exist in more than three dimensions (which would go far toward explaining its apparent movement).
Regardless, "essaying the Logrus" isn't just a matter of walking, though Merlin doesn't describe in more detail (that I recall) how one does negotiate the ever-changing path.
Unfortunately I do not have enough reputation to leave this as a comment, but I also can't find my copy of Manna From Heaven, so this is hardly a complete answer... but in it there is a quick prologue to Trumps of Doom (I believe) where Merlin goes through the Logrus and we see Frakir become animated. That would provide more details. If I find my copy I will update my answer.
UPDATE: Ah here we are. This is a section of the prologue [it's short!]. It leaves much to be desired, but as you can see, the suggestion of movement of the Logrus is there:
And the sensations of external movement had not abated. The floor of the tunnel seemed to ripple beneath his feet, the walls and roof to contract and expand. He stumbled, caught himself. Stumbled again...
At the next turning the sounds grew slightly louder, and he realized that they were not a tune, but rather a totally random concatenation of noises.
He climbed. He descended. The passageway shrank, and finally he crawled.
The sensations of movement increased. At times he seemed to be spinning; other times, it felt as if he were falling into an enormous abyss.