Ok, here's an expanded version of what I said in the comment.

## SNC case

First, I'm going to create a special log resolution of the pair $(X, X_0)$. We assume $d = \dim X$. We are assuming that $X$ is nonsingular and $X_0$ is simple normal crossings.

We begin by letting $S_{n}$ denote the stratum of this pair of dimension $n$. In other words, $S_0$ is the points that are intersections of components of $X_0$, $S_1$ is the curves that are components of $X_0$, etc. Let $Y^0 \to X$ be th blowup of all the points in $S_0$. This separates the strict transforms of the components of $S_1$. We let $S_{1}^0$ denote the set of those strict transforms (likewise with $S_{2}^0$, etc). Let $E_{0,i}^0 \subseteq Y_0$ denote the set of exceptional divisors.

Now, we can write
$$
K_{Y^0} = \pi_0^* K_X + (d-1) \sum E_{0,i}^0
$$
since we are blowing up nonsingular centers of codimension $d$ in a nonsingular variety. We also ahve
$$
\pi_0^* X_0 = (\pi_0)^{-1}_* X_0 + d \sum E_{0,i}^0.
$$
since each $E_{0,i}^0$ maps to a point in $S_0$ which is the intersection of $d$ components of $X_0$.

Next, we blowup the curves in $S_1^0$ (these are curves in $Y^0$). Note, each curve may intersect some $E_{0,i}^0$, but it cannot contain any, and likewise no $E_{0,i}^1$ contains any curve in $S_1^0$.. Let $E_{1, i}^1$ denote the exceptional divisors of this new blowup and $E_{0,i}^1$ the strict transforms of the old exceptional divisors. In this case we have
$$
K_{Y^1} = \pi_{1,0}^* K_{Y^0} + (d-2) \sum {E_{1,i}^1} = \pi_1^* K_X + (d-1) \sum E_{0,i}^1 + (d-2) \sum E_{1,i}^1.
$$
Here we crucially used the fact that no element of the the $S_1^0$ were contained in $E_{0,i}^0$.
By a similar consideration, we also have
$$
\pi_1^* X_0 = (\pi_1)^{-1}_* X_0 + d \sum E_{0,i}^1 + (d-1) E_{1,i}^1.
$$

Continuing in this way, we eventually obtain a log resolution that separates the components of $X_0$, $\pi : Y \to X$ and
$$
K_{Y} = \pi^* K_X + (d-1) \sum E_{0,i} + \dots + 1 \sum E_{d-2,i}
$$
where the $E_{0,i}$ are the strict transforms of the exceptional divisors from $Y^0 \to X$, $E_{1,i}$ are the strict transforms of the exceptional divisors from $Y^1 \to Y^0$, etc.
and also
$$
\pi^* X_0 = (\pi)^{-1}_* X_0 + d \sum E_{0,i} + (d-1) \sum E_{1,i} + \dots + 2 \sum E_{d-2, i}
$$
In other words for each $E$ an excpetional divisor on $Y$ (or an element of the strict transform), the coefficient of $\pi^* X_0$ is exactly one bigger than the coefficient in the relative canonical divisor (what you called Ram).

## SLC case

Let's now assume that

$X_0$ is semi-log canonical (SLC) but not necessarily SNC. In this case, since

$X_0$ is Gorenstein, this is equivalent to it being Du Bois (see work of Kovács and Doherty). It is also equivalent to the pair

$(X, X_0)$ being log canonical by inversion of adjunction.

Anyways, since $(X, X_0)$ is log canonical, for any resolution of singularities, $\pi : V \to X$, if we write
$$
K_V + D = \pi^* (K_X + X_0)
$$
we know that all the coefficients of $D$ are $\leq 1$ (this is the definition of log canonical). In other words, if we write
$$
K_V = \pi^* K_X + \mathrm{Ram}\;\;\;\; \text{ and }\;\;\;\; \pi^* X_0 = (\pi)^{-1}_* X_0 + \sum b_i E_i
$$
where $E_i$ are the exceptional divisors (note $\mathrm{Ram} = \sum c_i E_i$ for some $c_i$)
then we have that
$$
D = \pi^* K_X + \pi^* X_0 - K_V = \pi^* K_X + \pi^* X_0 - \pi^* K_X - \mathrm{Ram} = (\pi)^{-1}_* X_0 + \sum b_i E_i - sum c_i E_i.
$$
In other words, for each exceptional divisor $E_i$, we have that the coefficient of
$$
D = \pi^* X_0 - \mathrm{Ram}
$$
is $b_i - c_i$ and
$$
b_i - c_i \leq 1
$$
or in other words $b_i \leq c_i + 1$.

Note, we had equality in the SNC case when I picked a very special resolution of singularities.

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