$\newcommand\ga\gamma\newcommand\ze\zeta$

1. First of all, your operation $\circ$ has hardly anything to do with the Wasserstein metric. It is just an operation over measures over product spaces.

2. Assuming that $\ga_1$ is a probability measure over a product space $X\times Y$ (endowed with a corresponding product $\sigma$-algebra) and $\ga_2$ is a probability measure over a product space $Y\times Z$, let $(\xi,\eta)$ and $(\eta',\ze)$ be independent random elements of $X\times Y$ and $Y\times Z$, respectively (defined on the same probability space), with the respective distributions $\ga_1$ and $\ga_2$. Then for all appropriately measurable sets $A\subseteq X$ and $C\subseteq Z$, $$(\ga_2\circ\ga_1)(A\times C)=P(\xi\in A,\eta=\eta',\zeta\in C). \tag{1}$$ In my long career in probability, I have never seen probabilities like this, nor can I envision any use of them.

3. It is obvious from (1) that the measure $\ga_2\circ\ga_1$ will be a probability measure only in the trivial case when $P(\eta=\eta'=c)=1$ for some $c\in Y$.

4. The "composition" $\ga_2\circ\ga_1$ will be very unstable with respect to the distributions of the random elements $\eta$ and $\eta'$ of $Y$ -- a slight change in the distribution of either one of these two random elements may change the value of $(\ga_2\circ\ga_1)(A\times C)$ by as much as $1$.

5. As stated in the question you linked to your post, one can and does compose transition probabilities $P(x,B)$ and $Q(y,C)$: $$(QP)(x,C)=\int_Y P(x,dy)Q(y,C).$$ This does generalize the usual composition of functions. Indeed, the composition of (appropriately measurable) functions $f\colon X\to Y$ and $g\colon Y\to Z$ corresponds to the case when $P(x,B)=1(f(x)\in B)$ and $Q(y,C)=1(g(y)\in C)$, so that $(QP)(x,C)=1((g\circ f)(x)\in C)$.

$\newcommand\ga\gamma\newcommand\ze\zeta$

1. First of all, your operation $\circ$ has hardly anything to do with the Wasserstein metric. It is just an operation over measures over product spaces.

2. Assuming that $\ga_1$ is a probability measure over a product space $X\times Y$ (endowed with a corresponding product $\sigma$-algebra) and $\ga_2$ is a probability measure over a product space $Y\times Z$, let $(\xi,\eta)$ and $(\eta',\ze)$ be independent random elements of $X\times Y$ and $Y\times Z$, respectively (defined on the same probability space $(\Omega,\mathcal F,\mathsf P)$), with the respective distributions $\ga_1$ and $\ga_2$. Then for all appropriately measurable sets $A\subseteq X$ and $C\subseteq Z$, $$(\ga_2\circ\ga_1)(A\times C)=\mathsf P(\xi\in A,\eta=\eta',\zeta\in C). \tag{1}$$ In my long career in probability, I have never seen probabilities like this, nor can I envision any use of them.

3. It is obvious from (1) that the measure $\ga_2\circ\ga_1$ will be a probability measure only in the trivial case when $P(\eta=\eta'=c)=1$ for some $c\in Y$.

4. The "composition" $\ga_2\circ\ga_1$ will be very unstable with respect to the distributions of the random elements $\eta$ and $\eta'$ of $Y$ -- a slight change in the distribution of either one of these two random elements may change the value of $(\ga_2\circ\ga_1)(A\times C)$ by as much as $1$.

5. As stated in the question you linked to your post, one can and does compose transition probabilities $P(x,B)$ and $Q(y,C)$: $$(QP)(x,C)=\int_Y P(x,dy)Q(y,C).$$ This does generalize the usual composition of functions. Indeed, the composition of (appropriately measurable) functions $f\colon X\to Y$ and $g\colon Y\to Z$ corresponds to the case when $P(x,B)=1(f(x)\in B)$ and $Q(y,C)=1(g(y)\in C)$, so that $(QP)(x,C)=1((g\circ f)(x)\in C)$.

$\newcommand\ga\gamma\newcommand\ze\zeta$

1. First of all, your operation $\circ$ has hardly anything to do with the Wasserstein metric. It is just an operation over measures over product spaces.

2. Assuming that $\ga_1$ is a probability measure over a product space $X\times Y$ (endowed with a corresponding product $\sigma$-algebra) and $\ga_2$ is a probability measure over a product space $Y\times Z$, let $(\xi,\eta)$ and $(\eta',\ze)$ be independent random elements of $X\times Y$ and $Y\times Z$, respectively (defined on the same probability space), with the respective distributions $\ga_1$ and $\ga_2$. Then for all appropriately measurable sets $A\subseteq X$ and $C\subseteq Z$, $$(\ga_2\circ\ga_1)(A\times C)=P(\xi\in A,\eta=\eta',\zeta\in C). \tag{1}$$ In my long career in probability, I have never seen probabilities like this, nor can I envision any use of them.

3. It is obvious from (1) that the measure $\ga_2\circ\ga_1$ will be a probability measure only in the trivial case when $P(\eta=\eta'=c)=1$ for some $c\in Y$.

4. The "composition" $\ga_2\circ\ga_1$ will be very unstable with respect to the distributions of the random elements $\eta$ and $\eta'$ of $Y$ -- a slight change in the distribution of either one of these two random elements may change the value of $(\ga_2\circ\ga_1)(A\times C)$ by as much as $1$.

5. As stated in the question you linked to your post, one can and does compose transition probabilities $P(x,B)$ and $Q(y,C)$: $$(QP)(x,C)=\int_Y P(x,dy)Q(y,C).$$ This does generalize the usual composition of functions. Indeed, the composition of (appropriately measurable) functions $f\colon X\to Y$ and $g\colon Y\to Z$ corresponds to the case when $P(x,A)=1(f(x)\in A)$ and $Q(y,C)=1(g(y)\in C)$.

$\newcommand\ga\gamma\newcommand\ze\zeta$

1. First of all, your operation $\circ$ has hardly anything to do with the Wasserstein metric. It is just an operation over measures over product spaces.

2. Assuming that $\ga_1$ is a probability measure over a product space $X\times Y$ (endowed with a corresponding product $\sigma$-algebra) and $\ga_2$ is a probability measure over a product space $Y\times Z$, let $(\xi,\eta)$ and $(\eta',\ze)$ be independent random elements of $X\times Y$ and $Y\times Z$, respectively (defined on the same probability space), with the respective distributions $\ga_1$ and $\ga_2$. Then for all appropriately measurable sets $A\subseteq X$ and $C\subseteq Z$, $$(\ga_2\circ\ga_1)(A\times C)=P(\xi\in A,\eta=\eta',\zeta\in C). \tag{1}$$ In my long career in probability, I have never seen probabilities like this, nor can I envision any use of them.

3. It is obvious from (1) that the measure $\ga_2\circ\ga_1$ will be a probability measure only in the trivial case when $P(\eta=\eta'=c)=1$ for some $c\in Y$.

4. The "composition" $\ga_2\circ\ga_1$ will be very unstable with respect to the distributions of the random elements $\eta$ and $\eta'$ of $Y$ -- a slight change in the distribution of either one of these two random elements may change the value of $(\ga_2\circ\ga_1)(A\times C)$ by as much as $1$.

5. As stated in the question you linked to your post, one can and does compose transition probabilities $P(x,B)$ and $Q(y,C)$: $$(QP)(x,C)=\int_Y P(x,dy)Q(y,C).$$ This does generalize the usual composition of functions. Indeed, the composition of (appropriately measurable) functions $f\colon X\to Y$ and $g\colon Y\to Z$ corresponds to the case when $P(x,B)=1(f(x)\in B)$ and $Q(y,C)=1(g(y)\in C)$, so that $(QP)(x,C)=1((g\circ f)(x)\in C)$.