Consider two i.i.d. random sequences $(X_i)$ and $(Y_i)$. Assume that $$ \begin{aligned} \Pr[X_i=1]&=0.6 & \Pr[X_i=0]&=0.1 & \Pr[X_i=-1]&=0.3 \\ \Pr[Y_i=1]&=0.4 & \Pr[Y_i=0]&=0.4 & \Pr[X_i=-1]&=0.2~. \end{aligned}$$ Let $S_X(t)=\sum_{i=1}^tX_i$ and $S_Y(t)=\sum_{i=1}^tY_i$.
Then, for all $k>0$ and for all $t>0$, can we show $$\Pr[S_X(t)\geq k]\geq \Pr[S_Y(t)\geq k] \\ \text{or} \\ \Pr[S_X(t)\geq k]\leq \Pr[S_Y(t)\geq k] ~?$$
No, because for $t=1$ we get $$0.6=\Pr[S_X(1)\geq 1]>\Pr[S_Y(1)\geq 1]=0.4$$ but $$0.7=\Pr[S_X(1)\geq 0]<\Pr[S_Y(1)\geq 0]=0.8,$$ so which way round the inequality goes will depend on $k$. In fact you get a similar problem for every $t\geq 1$ by considering $k=t$ and $k=1-t$.
If the inequalities did all go the same way for $t=1$, that would actually imply the general case, since you could then find a coupling of the two sequences such that $X_i\geq Y_i$ for every $i$ (or vice versa).