How do the poems "Mutability" and "Ozymandias" use the ironic conflict between being (stasis, fixity, imaginative death) and becoming (process, change, progress) to explore the problem of mutability?

Mutability is defined as the tendency or liability to change. Something that is mutable is, at its core, incapable of establishing a sense of regularity or identity. As such, something cannot both be and become at the same time. This is the ironic tension described in your question.
This tension is at play in Percy Bysshe Shelley’s “Mutability” and “Ozymandias.”
In “Mutability,” the first three stanzas describe examples of the ephemeral. Shelley explains how once something occurs, something else occurs that replaces the original. Something can never be replicated identically once it has passed. This relates to the state of being and becoming because in Shelley’s view, things just are until they are not. In the final stanza, Shelley underscores his message that human existence is impermanent when he says, “Man's yesterday may ne'er be like his morrow; / Nought may endure” except change.
In “Ozymandias,” the speaker describes a fallen statue in the desert of what was likely a powerful man during his lifetime. The inscription on the statue warns the viewer to “Look on my works, Ye mighty, and despair!” This contrasts with the statue’s destruction and isolation. The viewer has no works upon which he or she can look to judge Ozymandias’s prowess as a ruler. This relates to the concept of mutability because it is clear that the statue has changed from its original state. When Ozymandias commissioned the statue, he obviously believed it would tower over his kingdom for the rest of time. Of course, nothing lasts forever, and Shelley uses the ego of a fallen ruler to demonstrate that. When the statue came into being, it probably enjoyed fixity for quite some time. Despite this period, the statue eventually changed—along with its subject’s influence on the land and history.
In both poems, Shelley philosophizes on the nature of time and permanence and reaches the conclusion that the world—and everything in it—is in a constant state of flux.