What are the specific assumptions of MRS? I know MRS is independent of prices and compares the ratio of goods which the consumer will exchange one good for the other good. But is the marginal rate of substitution affected by the consumption of those goods or not? It seems trivial but can't get my head around it.
Different consumer preferences will lead to different properties of the consumer's willingness to trade one good for another.

For instance, suppose that consumers preferences are linear in the consumption of either good:
$U=\beta_x x + \beta_y y$
where $\beta_x$,$\beta_y$ are positive constants 

The marginal utility of $x$ would be $\beta_x$ and the marginal utility of $y$ would be $\beta_y$, and the MRS would be $\frac{\beta_x}{\beta_y}$, also a constant. This tells us the consumer would be willing to trade between $x$ and $y$ at a constant rate.

An alternative would be consumption of each good entering utility in a multiplicative fashion (Cobb Douglas), for instance
$U=xy$. The marginal utility of $x$ would be $y$ and the marginal utility of $y$ would be $x$ and so the MRS would be $\frac{y}{x}$, which indeed would depend on consumption. In this example, the MRS decreases as consumption of $x$ rises. As the consumer enjoys more $x$ he is willing to give up less and less $y$ for another unit of $x$.

So it's not entirely trivial whether the MRS depends on consumption, whether it does or not depends on how you model consumer's preferences.