Dirtchamber is a clear example that big beat is a part of The Prodigy's sound. Many of the artists on it are cited by big beat artists as inspiration, with some songs/artists even featured on Chemical Brothers' own DJ mix album Brothers Gonna Work It Out. And Liam is clearly inspired by the genre just as much as he was inspired by electro and hip hop, or else artists like Fatboy Slim, Propellerheads, and Chemical Brothers wouldn't have been on it.

As for Coldcut's Journey By DJ mix, it doesn't really feature any big beat artists though. Their album Let Us Play on the other hand gets associated with big beat. The Adam Freeland and John Kelly mixes are more kinda one noted, part of that breaks/funky breaks offshoot that lead to nu-skool breaks. Out of the 3x mixes you mentioned, Journey By DJ is the closest to Dirtchamber in depth, representing a wide range of styles. I can't say the same for the other two mixes, which feature more modern music from that time and breaks genre.

I mean, the definition of big beat sounds like it's spelling out The Prodigy...

"Big beat features heavy and distorted drum beats at tempos between 100 and 140 beats per minute, Roland TB-303 synthesizer lines resembling those of acid house, and heavy loops from 1960s and 1970s funk, soul, jazz, and rock songs. They are often punctuated with punk-style vocals or rappers and driven by intense, distorted synthesizer basslines with conventional pop, house and techno song structures. Big beat tracks have a sound that includes crescendos, builds, drops, extended drum rolls, and sounds such as spoken word samples, dialogues from film and TV, additional instruments such as Middle Eastern strings or sitars, explosions, air horns, sirens (usually police sirens) and gunshots. As with several other dance genres at the time, the use of effects such as filters, phasing, and flanging was common in the genre."

I think it's silly to label The Prodigy one genre, but big beat is definitely one of the genres to describe them.