Let $\mu_k$ and $\nu_k$ denote the $k$th moments of $\mu$ and $\nu$ respectively.

Write $$W_p(\mu,\nu)=\inf\{\|X-Y\|_p\colon X\sim\mu,Y\sim\nu\},$$ where $\|Z\|_p:=(E|Z|^p)^{1/p}$.

Then for any random variables $X$ and $Y$ such that $X\sim\mu$ and $Y\sim\nu$ we have $$|\mu_2^{1/2}-\nu_2^{1/2}|=|\|X\|_2-\|Y\|_2|\le\|X-Y\|_2$$ and $$|\mu_1-\nu_1|=|EX-EY|\le\|X-Y\|_1\le\|X-Y\|_2.$$

So, $$|\mu_2^{1/2}-\nu_2^{1/2}|\le W_2(\mu,\nu)$$ and $$|\mu_1-\nu_1|\le W_2(\mu,\nu).$$

Let $\mu_k$ and $\nu_k$ denote the $k$th moments of $\mu$ and $\nu$ respectively.

Write $$W_p(\mu,\nu)=\inf\{\|X-Y\|_p\colon X\sim\mu,Y\sim\nu\},$$ where $\|Z\|_p:=(E|Z|^p)^{1/p}$.

$\newcommand\si\sigma$Then for any random variables $X$ and $Y$ such that $X\sim\mu$ and $Y\sim\nu$ we have $$|\mu_2^{1/2}-\nu_2^{1/2}|=|\|X\|_2-\|Y\|_2|\le\|X-Y\|_2$$ and $$|\mu_1-\nu_1|=|EX-EY|\le\|X-Y\|_1\le\|X-Y\|_2.$$

So, $$|\mu_2^{1/2}-\nu_2^{1/2}|\le W_2(\mu,\nu) \tag{1}\label{1}$$ and $$|\mu_1-\nu_1|\le W_2(\mu,\nu).$$

Concerning \eqref{1}, note that $|\mu_2-\nu_2|$ cannot be bounded in terms of $W_2(\mu,\nu)$. Indeed, if e.g. $\mu$ and $\nu$ are the centered normal distributions over $\Bbb R$ with respective standard deviations $\si+1$ and $\si$, then, by Proposition 7, $W_2(\mu,\nu)=1$, whereas $|\mu_2-\nu_2|=2\si+1\to\infty$ as $\si\to\infty$.