Bounds on difference of expectations vs expectation of difference

For random variables $X$, $Y$ with finite mean, I am interested in studying the difference between $\mathbb{E}[|X-Y|]$ and $\left|\mathbb{E}[X]-\mathbb E[Y]\right|$.

If $X$, $Y$ have equal mean then $\left|\mathbb{E}[X]-\mathbb E[Y]\right|=0$, while $\mathbb{E}[|X-Y|]>0$ as long as they are not both Dirac distributions. Does it always hold that $\left|\mathbb{E}[X]-\mathbb E[Y]\right|\leq \mathbb{E}[|X-Y|]$?

I am interested in a positive or negative answer to the above, as well as any information on how I can bound the difference.

For any integral against a positive measure, we have: $$\left| \int f d\mu \right| \leq \int |f| d\mu$$

If we set $\mu = \mathbb{P}$, then we get by linearity of integration/expectation: $$|\mathbb{E}(X) - \mathbb{E}(Y)| = |\mathbb{E}(X-Y)| \leq \mathbb{E}|X-Y|$$