Good upper-bound for $\sup_t \|F_t(A)-F_t(B)\|_{op}$ where $F_t(A) := (I-\eta A)^t$ and $\|A-B\|_F \le \epsilon$

Question

Let $A$ and $B$ be psd matrices such that $\|A-B\|_F \le \epsilon$. Given an integer $t \ge 0$ and sufficiently small positive $\eta$, define $F_t(A) := (I_d-\eta A)^t$ and $K_t(A) := \sum_{0 \le s \le t-1}F_s(A)$.

Q: What are good upper-bounds for $\sup_t\|F_t(A)-F_t(B)\|_{op}$ and $\sup_t \|K_t(A)-K_t(A)\|_{op}$ ?