Let us examine what happens when we price our bread and butter, the vanilla interest rate swap in two worlds - the single curve world and the multi curve world.
Let the first reset date be $T_\alpha$ and the last payment date be $T_\beta$.
In the single curve world, the vanilla IRS has PV at time $t$ to be $$ \begin{align} \pi_t & = \mathbb{E}^{ \mathbb{Q} }_{t} \left[ \sum_{i} D_{tT_i} \tau_i \left[ L(T_{i-1};T_{i-1},T_i) - K \right] \right] \\ & = \sum_{i} P_{tT_i} \tau_i \left[ \mathbb{E}^{ \mathbb{Q}^{T_i} }_{t} \left[ L(T_{i-1};T_{i-1},T_i) \right] - K \right] \\ & = \sum_{i} P_{tT_i} \tau_i \left[ L(t;T_{i-1},T_i) - K \right] \\ & = \sum_{i} P_{tT_i} \tau_i L(t;T_{i-1},T_i) - K \sum_{i} P_{tT_i} \tau_i \\ & = \sum_{i} P_{tT_i} \tau_i \frac{1}{\tau_i} \left[ \frac{P_{tT_{i-1}} }{P_{tT_i}} -1 \right] - K \sum_{i} P_{tT_i} \tau_i \\ & = \sum_{i} P_{tT_i} \left[ \frac{P_{tT_{i-1}} }{P_{tT_i}} -1 \right] - K \sum_{i} P_{tT_i} \tau_i \\ & = P_{tT_\alpha} - P_{tT_\beta}-K \sum_{i} P_{tT_i} \tau_i \end{align} $$
In the multi curve world, the vanilla IRS has PV at time $t$ to be
$$ \begin{align} \pi_t & = \mathbb{E}^{ \mathbb{Q} }_{t} \left[ \sum_{i} D^{\text{ois}}_{tT_i} \tau^{\text{ois}}_i \left[ L(T_{i-1};T_{i-1},T_i) - K \right] \right] \\ & = \sum_{i} P^{\text{ois}}_{tT_i} \tau^{\text{ois}}_i \left[ \mathbb{E}^{ \mathbb{Q}^{T_i} }_{t} \left[ L(T_{i-1};T_{i-1},T_i) \right] - K \right] \\ & = \sum_{i} P^{\text{ois}}_{tT_i} \tau^{\text{ois}}_i \left[ L(t;T_{i-1},T_i) - K \right] \\ & = \sum_{i} P^{\text{ois}}_{tT_i} \tau^{\text{ois}}_i L(t;T_{i-1},T_i) - K \sum_{i} P^{\text{ois}}_{tT_i} \tau^{\text{ois}}_i \\ & = \sum_{i} P^{\text{ois}}_{tT_i} \tau^{\text{ois}}_i \frac{1}{\tau_i} \left[ \frac{P_{tT_{i-1}} }{P_{tT_i}} -1 \right] - K \sum_{i} P^{\text{ois}}_{tT_i} \tau^{\text{ois}}_i \\ \end{align} $$ Setting $\pi_t=0$, i.e., entering into the swap at time $t$ has no cost, means that the swap rate is $$ K=\frac{\sum_{i} P^{\text{ois}}_{tT_i} \tau^{\text{ois}}_i \frac{1}{\tau_i} \left[ \frac{P_{tT_{i-1}} }{P_{tT_i}} -1 \right] }{\sum_{i} P^{\text{ois}}_{tT_i} \tau^{\text{ois}}_i} $$
The difference is that now both ZCB curves are required to value the swap. The risk neutral measure $\mathbb{Q}$ is now explicitly under the discounting curve. You still assume that the projection curve is a martingale under $\mathbb{Q}$, though.
