The local volatility graph tomorrow doesn't change, unless the implied volatility surface tomorrow is not the same as today. LV takes the implied vol surf today as input, and outputs a instantaneous volatility function of spot and time, which can price vanilla options today exactly the same as the market prices.
In local volatility world, you assume the implied vol surf does not change. If there is a change in the implied vol surface tomorrow, you have to remark the implied volatilities, and create a new local vol graph, which is alike to remarking implied vol in Black Scholes. If you do not remark, you will have un-explained P&L, because the old local vol function can no longer perfectly calibrate to your vanillas options anymore, which gives you wrong vanilla prices and un-explained P&L.
In your picture, you only have one red line for implied volatility, which i assume your implied vol surface at $t=1$ is same as $t=0$.
Also assuming $dt$ is small,
$$\sigma_{LV}(t=1,spot=S_1)=Lower\ one\ of\ "LV(t=1)?"$$
Loosely speaking, implied variance is the average integrated local variance from today up to time T weighted by dollar gamma. For more, you should get familiar with skew stickiness ratio (ratio of ATMF instantaneous vol's slope over log spot over ATMF implied vol's slope over log strike) by Bergomi, which tells you that $SSR_{LV}=2$
"And since LV doesn't vary with strike"
- It's because LV describes the instantaneous vol of the spot, not the implied vol of any option.
"LV model the volatility is constant through different strikes"
Not sure if i understood correctly. What volatility are you referring to?
If you are saying the instantaneous vol of the spot, the function $\sigma_{LV}(S,t)$ does not depend on strike, since it describe the spot price's instantaneous volatility not an option's implied vol
If you are saying the evolution of implied vol of an option ($K, T$) under LV dynamics, it is a function of spot price and time $t$
$$\hat\sigma_{K,T}(S,t)=C_{BS}^{(-1)}(S,K,T-t)=f(S,t|\Sigma)$$, where spot dynamics is
$$dS=(r-q)Sdt+\sigma(S,t)SdW$$
- By Ito's lemma, $$d\hat\sigma_{K,T}=\frac{\partial f}{\partial S}dS+\frac{\partial f}{\partial t}dt+\frac{1}{2}\frac{\partial^2 f}{\partial S^2}dS^2$$
$$d\hat\sigma_{K,T}=\frac{\partial f}{\partial S}[(r-q)Sdt+\sigma(S,t)SdW]+\frac{\partial f}{\partial t}dt+\frac{1}{2}\frac{\partial^2 f}{\partial S^2}\sigma(S,t)^2S^2dt$$
$$d\hat\sigma_{K,T}=[\frac{\partial f}{\partial S}(r-q)S+\frac{\partial f}{\partial t}+\frac{1}{2}\frac{\partial^2 f}{\partial S^2}\sigma(S,t)^2S^2]dt + \frac{\partial f}{\partial S}\sigma(S,t)SdW$$
