Does an injective map stay injective under small smooth perturbations?

There is a standard trick for dealing with this. If your parameter $t$ is in some interval $I$, consider $F\colon M\times I\to N\times I$ given by $$F(x,t) = (f_t(x),t).$$ Assuming that $f=f_0$ is an immersion, we see that $F$ is an immersion at $(x,0)$ for all $x\in M$, and hence one-to-one in a neighborhood of $(x,0)$. Therefore, for large enough $n$, $F(x_n,t_n)=F(y_n,t_n)$ forces $x_n=y_n$, completing your contradiction.