  In this paper, we establish the monotone convex order between two
$\mathbb{R}$-valued McKean-Vlasov processes $X=(X_t)_{t\in [0, T]}$ and
$Y=(Y_t)_{t\in [0, T]}$ defined on a filtered probability space $(\Omega,
\mathcal{F}, (\mathcal{F}_{t})_{t\geq0}, \mathbb{P})$ by \begin{align}
&dX_{t}=b(t, X_{t}, \mu_{t})dt+\sigma(t, X_{t}, \mu_{t})dB_{t}, \quad X_{0}\in
L^{p}(\mathbb{P})\; \text{with}\; p\geq 2,\nonumber\\ &dY_{t}=\beta(t, Y_{t},
\nu_{t})dt+\theta(t, \,Y_{t}, \nu_{t})\,dB_{t}, \,\quad Y_{0}\in
L^{p}(\mathbb{P}), \nonumber \end{align} where $\forall\, t\in [0, T],\:
\mu_{t}=\mathbb{P}\circ X_{t}^{-1}, \:\nu_{t}=\mathbb{P}\circ Y_{t}^{-1}. $ If
we make the convexity and monotony assumption (only) on $b$ and $|\sigma|$ and
if $b\leq \beta$ and $|\sigma|\leq |\theta|$, then the monotone convex order
for the initial random variable $X_0\preceq_{\,\text{mcv}} Y_0$ can be
propagated to the whole path of processes $X$ and $Y$. That is, if we consider
a non-decreasing convex functional $F$ defined on the path space with
polynomial growth, we have $\mathbb{E}\, F(X)\leq \mathbb{E}\, F(Y)$; for a
non-decreasing convex functional $G$ defined on the product space involving the
path space and its marginal distribution space, we have $\mathbb{E}\, G(X,
(\mu_{t})_{t\in [0, T]})\leq \mathbb{E}\, G(Y, (\nu_{t})_{t\in [0, T]})$ under
appropriate conditions. The symmetric setting is also valid, that is, if
$Y_0\preceq_{\,\text{mcv}} X_0$ and $|\theta|\leq |\sigma|$, then $\mathbb{E}\,
F(Y)\leq \mathbb{E}\, F(X)$ and $\mathbb{E}\, G(Y, (\nu_{t})_{t\in [0, T]})\leq
\mathbb{E}\, G(X, (\mu_{t})_{t\in [0, T]})$. The proof is based on several
forward and backward dynamic programming principle and the convergence of the
truncated Euler scheme of the McKean-Vlasov equation.
