Number of rounds for constant header size in common hashes and XOFs We compute hash $H(M_0\mathbin\|M_1)$ of size $d\ge1$ for some constant header $M_0$ of size $m_0$, and $\nu\ge1$ messages $M_1$ of random content and size $m_1$.
For Merkle-Damgård hashes, a simple optimization pre-computes the first $\left\lfloor m_0/r\right\rfloor$ rounds (where $r$ is the block size), and for some minimum extra size $\mu$, the total number of rounds computed is
$$\text{rounds}=\left\lfloor\frac{m_0}r\right\rfloor+\nu\left\lceil\frac{\left(m_0\bmod r\right)+m_1+\mu}r\right\rceil$$
with, for some hashes:
$$\begin{array}{l|rrrl|rrrl}
\text{hash} &amp; d&amp;r&amp;\mu&amp;\text{(bits)}&amp;\tilde d&amp;\tilde r&amp;\tilde \mu&amp;\text{(bytes)}\\
\hline
\operatorname{SHA-256}    &amp; 256 &amp;  512 &amp;  65 &amp;&amp; 32 &amp;  64 &amp;  9 \\
\operatorname{SHA-512}    &amp; 512 &amp; 1024 &amp; 129 &amp;&amp; 64 &amp; 128 &amp; 17 \\
\end{array}$$
Note: The right side of the table assumes quantities are expressed in byte rather than bit. The only change to the formula is that variables get a tilde, except $\nu$.
For more common hashes and XOFs¹, what are the corresponding formulas and parameters?
I'm especially interested in common variants of SHA-3 (or KECCAK if it differs from SHA-3 other than by value of padding); SHAKE; various BLAKE; and perhaps some parallelizable hashes.
Update: I have changed notation several times to align with that in KECCAK/SHAKE for $r$ and output size $d$, and used greek letters in an attempt to avoid confusion with existing hash specifications.

¹ eXtendable Output Functions essentially are hashes where the output size $d$ is a parameter rather than fixed. An example is SHAKE256 of the SHA-3 family.
Heading
\begin{array}{l|rrrl|rrrl}
\text{hash} &amp; d&amp;r&amp;\mu&amp;\text{(bits)}&amp;\tilde d&amp;\tilde r&amp;\tilde 
 \mu&amp;\text{(bytes)}\\
\hline
\operatorname{MD5}        &amp; 128 &amp;  512 &amp;  65 &amp;&amp; 16 &amp;  64 &amp;  9 \\
\operatorname{SHA-1}      &amp; 160 &amp;  512 &amp;  65 &amp;&amp; 20 &amp;  64 &amp;  9 \\
\operatorname{RIPEMD-160} &amp; 160 &amp;  512 &amp;  65 &amp;&amp; 20 &amp;  64 &amp;  9 \\
\operatorname{SHA-224}    &amp; 224 &amp;  512 &amp;  65 &amp;&amp; 28 &amp;  64 &amp;  9 \\
\operatorname{SHA-256}    &amp; 256 &amp;  512 &amp;  65 &amp;&amp; 32 &amp;  64 &amp;  9 \\
\hline
\operatorname{SHA-512/224}&amp; 224 &amp; 1024 &amp; 129 &amp;&amp; 28 &amp; 128 &amp; 17 \\
\operatorname{SHA-512/256}&amp; 256 &amp; 1024 &amp; 129 &amp;&amp; 32 &amp; 128 &amp; 17 \\
\operatorname{SHA-384}    &amp; 384 &amp; 1024 &amp; 129 &amp;&amp; 48 &amp; 128 &amp; 17 \\
\operatorname{SHA-512}    &amp; 512 &amp; 1024 &amp; 129 &amp;&amp; 64 &amp; 128 &amp; 17 \\
\hline
\operatorname{SHA3-224}  &amp; 224 &amp;  1152 &amp;  4 &amp;&amp; 28  &amp; 144 &amp;  1 \\
\operatorname{SHA3-256}  &amp; 256 &amp;  1088 &amp;  4 &amp;&amp; 32  &amp; 136 &amp;  1 \\
\operatorname{SHA3-384}  &amp; 384 &amp;   832 &amp;  4 &amp;&amp; 48  &amp; 104 &amp;  1 \\
\operatorname{SHA3-512}  &amp; 512 &amp;   576 &amp;  4 &amp;&amp; 54  &amp; 72 &amp;  1 \\
\hline
\operatorname{SHAKE-128} &amp; d   &amp;  1344 &amp;  6 &amp;&amp; \lceil d/8 \rceil &amp; 168 &amp;  1 \\
\operatorname{SHAKE-256} &amp; d   &amp;  1088 &amp;  6 &amp;&amp; \lceil d/8 \rceil &amp; 136 &amp;  1 \\
\hline
\operatorname{BLAKE2s-256}   &amp; 256 &amp;   512 &amp; 0 &amp;&amp; 32 &amp; 64 &amp;  0 \\
\operatorname{BLAKE2b-512}   &amp; 512 &amp;  1024 &amp; 0 &amp;&amp; 64 &amp; 128 &amp; 0  \\
\end{array}

SHA3-x
SHA3-x is defined with KECCAK as;

$\operatorname{SHA3-224}(M) = \operatorname{KECCAK}[448] (M \mathbin\| 01, 224)$
$\operatorname{SHA3-256}(M) = \operatorname{KECCAK}[512] (M \mathbin\| 01, 256)$
$\operatorname{SHA3-384}(M) = \operatorname{KECCAK}[768] (M \mathbin\| 01, 384)$
$\operatorname{SHA3-512}(M) = \operatorname{KECCAK}[1024](M \mathbin\| 01, 512)$

KECCAK
and KECCAK defined as

$\operatorname{KECCAK}[c] (N, d) = \operatorname{SPONGE}[\operatorname{KECCAK-p}[1600, 24], \operatorname{pad10*1}, 1600–c] (N, d)$

and note that $N = M\mathbin\|01$ and $d$ is the required output size, Now we have

$\operatorname{SHA3-224}(M) = \operatorname{SPONGE}[\operatorname{KECCAK-p}[1600, 24], \operatorname{pad10*1}, 1600–448] (N, 224)$
$\operatorname{SHA3-256}(M) = \operatorname{SPONGE}[\operatorname{KECCAK-p}[1600, 24], \operatorname{pad10*1}, 1600–512] (N, 256)$
$\operatorname{SHA3-384}(M) = \operatorname{SPONGE}[\operatorname{KECCAK-p}[1600, 24], \operatorname{pad10*1}, 1600–768] (N, 384)$
$\operatorname{SHA3-512}(M) = \operatorname{SPONGE}[\operatorname{KECCAK-p}[1600, 24], \operatorname{pad10*1}, 1600–1024] (N, 512)$

Padding

$\operatorname{pad10*1}(x, m)$  is not important since the message is not formed yet.

SPONGE

$\operatorname{SPONGE}[f, \operatorname{pad}, r](N, d)$

Let $P=N \mathbin\| \operatorname{pad}(r, \operatorname{len}(N))$.

Let $n=\lfloor\operatorname{len}(P)/r\rfloor$.

Let $c=\lfloor b/r\rfloor$.

Let $P_0, \ldots, P_{n-1}$ be the unique sequence of strings of length $r$ such that $P = P_0 \mathbin\| \ldots\mathbin\| P{n-1}$.

Let $S=0^b$

$\textbf{For } i \textbf{ from }0 \textbf{ to } n-1$
$\textbf{let } S=f (S \oplus (P_i\mathbin\| 0c))$.

...




As we can see, we can precompute a message $M$ up to the maximum multiple of $r$  that is smaller than $\operatorname{len}(M)$. More mathematically $$\operatorname{precomputableLen} = \lfloor(\operatorname{len}(M)/r)\rfloor \cdot r.$$ And the $r$ for

$\operatorname{SHA3-224}(M)$ is $r = 1152$
$\operatorname{SHA3-256}(M)$ is $r = 1088$
$\operatorname{SHA3-384}(M)$ is $r = 832$
$\operatorname{SHA3-512}(M)$ is $r = 576$

and the formula
$$\text{rounds}=\left\lfloor\frac{m_0}r\right\rfloor+\nu\left\lceil\frac{\left(m_0\bmod r\right)+m_1+\mu }r\right\rceil$$
SHAKE128 and SHAKE256

$\operatorname{SHAKE128}(M, d) = \operatorname{KECCAK}[256] (M \mathbin\| 1111, d)$
$\operatorname{SHAKE256}(M, d) = \operatorname{KECCAK}[512] (M \mathbin\| 1111, d)$

As we can see the main difference is the capacity and extra two bits appended to have the domain separation.

$\operatorname{SHAKE128}(M, d)$ is $r =1344$
$\operatorname{SHAKE256}(M, d)$ is $r =1088$

and the formula
$$\text{rounds}= \underbrace{\left\lfloor\frac{m_0}{r}\right\rfloor + \nu\left\lceil\frac{\left(m_0\bmod r\right)+m_1+\mu}r\right\rceil}_{\text{Absorbing part}} + \underbrace{\nu  \left\lceil \frac{d}{r} -1 \right\rceil  }_{\text{Squeezing part}}$$
BLAKE2b and BLAKE2s
BLAKE2 uses modified ChaCha as compression function with 16 words. BLAKE2s is the 32-bit version so  $r=512$ here, and for BLAKE2b $r = 1024$ ( $s$ for small, $b$ for big).
and the formula
$$\text{rounds}=\left\lfloor\frac{m_0}r\right\rfloor+\nu\left\lceil\frac{\left(m_0\bmod r\right)+m_1+\mu } r\right\rceil$$
since BLAKE2 uses all-zero padding, they called it minimal padding.