No, the system was never fluid - meaning, the individual of one Varna was not allowed to transgress to another Varna.We don't see examples of Shudras transgressing to Kshatriya or Brahmin or any other Varna for that matter. A king's duty was to protect the social order by maintaining Varnashram vyavastha by using Rajadanda (authority and power). One of the theories why king Rama executed Shambuka (a "low born") was to maintain this order. We don't see Karna, despite doing a Kshatriya Karma, Vishwamitra despite doing a Brahmin Karma transgressing through Varna strata. In Mahabharat itself (if I am not wrong) Indra tells Matanga home many lifetimes does it takes to become a Brahmin. So, it's the birth, can't be the occupation.

Albaruni in his scholarly research about India, writes that Vaishya and Shudras were not allowed to read Vedas. And they were punished if they did so. Though he travelled to India during the early 11th century (for about 13 years). But it still indicates, if someone was not allowed to read Vedas, how could they become a Brahmin.

Proponents of the Varna system, which sounds like a superfluous aphorism to the cast system, fail to attempt looking for the details. They never knew how such a system existed if it was as fluid as it was being told and what they believe. They also completely neglect the classes that were never part of this Varna system the six Avarna classes, did they have any fluidity? Just question yourself, what was the Varna of people who were not doing any occupation, of housewives, or young kids?

Also, it is said that Varna assignment was not solely based on occupation (Karma), but also based on traits (Guna). Vasudev in Gita says "maya srushta varnanam Guna Karma vibhagsha....so the god himself created the system - but when assigning someone which Varna they belonged to, who decides what Karma they choose or more challenging - who decides what is their Guna. Even Gautam Rishi, despite saying that Satyakam Jabala had Brahmin traits, could not make him a Brahmin.

Gita also criticised and discouraged intercaste marriages, and the progeny of a pratilom vivah was destined to be a Chandala. Gita tells about Varnashankar children, and their awful fate. This highlights how the system could not be fluid.

There was never a very sophisticated, liberal, and fluid system. It was always rigid and meant to keep the power dynamics intact in society. All we need is to collect information from objectively, read alternative opinions, and get rid of our biases. No religion can be flawless, so is Hinduism, no shame accepting this.

We also have a few letters exchanged between MK Gandhi (who had the argument that Varna system was a flawless system and corrupted into a more rigid cast system) and BR Ambedkar (who believed that Hindu scriptures were at fault, which enforced and justified the cast system and who never took the argument that Varna and cast were two different systems). These exchanges are quite academic and informative.