Existence of projection $P$ equivalent to $P' \circ T \circ P''$ for projections $P', P''$ and smooth translation $T$?

Question

A projection in the linear algebraic sense is a linear map $P$ such that $P^2 = P$. I'm interested in knowing when there is guaranteed to exist a projection $P$ such that $P = P' \circ T \circ P''$, where $P', P''$ are two arbitrary projections, and $T$ represents translation by a smooth curve (for instance, moving along the path $\gamma(x) = x^2, x \in [0,1]$, though I am using "smooth translation" in a loose sense, so would be open to various formalizations in answers). I am working in (high-dimensional) Euclidean space, and the motivation for this is $T$ represents a gradient flow in a machine learning settings, and the projections represent dimensionality reduction during the gradient flow. But I am interested in the general mathematical answer for when we know or believe such a projection $P$ is guaranteed to exist. For instance, this could be when we have certain conditions on any of $P', T, P''$, or on the ambient space or gradient flow.