Between the Schrödinger and Heisenberg picture, the latter is the one most directly connected to dynamics in the form you're used to seeing it in, with the dynamic equations governing the various quantities; be they ordinary differential equations of motion or partial differential equations for fields. In quantum theory, in the Heisenberg Picture, those same equations hold for the quantized versions of the same - up to operator ordering ambiguity, while the state becomes a timeless denotation of an entire history, rather than that of a system and its progression in time.
To answer your question in the most direct way possible, it's first necessary to clarify that the wave function vector $|ψ❭$ is not the state, but is better regarded as a “square root” of the state. Among other issues, it has both phase and normalization ambiguity: different non-zero rescalings and different phases yield the same state. The best way to handle that and remove the ambiguity is to refer, instead, to $W = |ψ❭❬ψ|/❬ψ|ψ❭$ as the state.
These are pure states. A mixture of such states corresponding to mutually orthogonal vectors, with non-negative mixing coefficients that add up to $1$ gives you a mixed state. More general mixed states can be contemplated that are continuous linear combinations of pure states, rather than discrete sums. Example: $W_C = p |0❭❬0| + q |1❭❬1|$ is a classical bit (a mixed state), while
$$W_Q = \left(\sqrt{p} |0❭ + \sqrt{q} e^{iφ} |1❭\right)\left(\sqrt{p} ❬0| + \sqrt{q} e^{iφ} ❬1|\right) = W_C + \sqrt{pq} \left(e^{iφ} |0❭❬1| + e^{-iφ} |1❭❬0|\right)$$
is a quantum bit, which is a pure state. The state $½ W_C + ½ W_Q$ would then be $50%$ pure; so purity can range from $0%$ to $100%$.
A physical quantity $A$, when quantized, becomes an operator $\hat{A}$ that has a decomposition of the form $\hat{A} = \sum_a a P_a$, into a linear combination of projection operators $P_a$ and eigenvalues $a$. The effect of the projection $P_a|ψ❭$ is to reduce $|ψ❭$ to the sum of its a-eigenvector components only; zeroing out all the other eigenvectors - it projects $|ψ❭$ down to the eigenspace of the value $(a)$ for operator $\hat{A}$. Since the eigensubspaces span the entire space for $|ψ❭$, then it is also assumed the projections add to $1$: $\sum_a P_a = 1$.
More general operators can be considered that have spectra that are continuous, or mixed continuous/discrete. It won't shed any light to consider them here, so I'll just keep to the simpler case of discrete spectra only.
The Born rule says that the result of applying the measurement for quantity $A$ is a quantum event that reduces the state $|ψ❭$ to the state $P_a|ψ❭$ with probability $\left|P_a|ψ❭\right|^2/❬ψ|ψ❭$ and that the value measured by the event is $(a)$.
When restated, the rule asserts that the state $W = |ψ❭❬ψ|/❬ψ|ψ❭$ becomes
$$\frac{P_a|ψ❭❬ψ|{P_a}^†}{❬ψ|{P_a}^†P_a|ψ❭} = \frac{P_a|ψ❭❬ψ|P_a}{❬ψ|P_a|ψ❭}$$
... the latter equality applying since ${P_a}^† = P_a = {P_a}^2$ for projection operators. It states that this happens with probability $|P_a|ψ❭|^2/❬ψ|ψ❭ = ❬ψ|P_a|ψ❭/❬ψ|ψ❭$.
Introducing the “trace” operator, defining it by the property $\text{Tr}(A|ψ❭❬ψ|B) = ❬ψ|BA|ψ❭$, then the above reduction can be restated as $W → P_a W P_a/\text{Tr}\left(W P_a\right)$ with probability $\text{Tr}\left(W P_a\right)$.
There are two ways to treat this, depending on what you consider a mixed state to represent. If a mixed state $W = p W_0 + q W_1$ ($p,q ≥ 0$, $p + q = 1$) stands for “$W_0$ with probability $p$, $W_1$ with probability $q$” (recursively applied to $W_0$ and $W_1$ if they are also mixed states) then the entire reduction itself can be succinctly wrapped up as:
$$W → W_A ≡ \sum_a \text{Tr}\left(W P_a\right) × \frac{P_a W P_a}{\text{Tr}\left(W P_a\right)} = \sum_a P_a W P_a$$.
Thus, each quantum event produces a change of the form $W → W_A = \sum_a P_a W P_a$, where $A$ is the corresponding quantity that is being measured by that event.
Note that this normalizes correctly since
$$\text{Tr}\left(W_A\right) = \sum_a \text{Tr}\left(W P_a\right) = \text{Tr}\left(W \sum_a P_a\right) = \text{Tr}(W) = 1.$$
The other way to regard a mixed state as being a thing in its own right, so that the transition has two steps: the first producing the mixed state itself, and the second producing one of its pure state components with the associated probability. Personally, I think that way of looking at it introduces unnecessary redundancy, so I will keep to the first interpretation.
If a subsequent measurement is made for a second quantity $C$ whose quantized form $\hat{C}$ decomposes into $\hat{C} = \sum_c {P'}_c$, then the result will be a reduction to the state
$$W_A → W_{AC} = \sum_c {P'}_c W_A {P'}_c = \sum_{a,c} {P'}_c P_a W P_a {P'}_c.$$
This can be wrapped up by introducing the time-ordering pseudo-operator $T[\_]$ and its dual $T'[\_]$, defined with the properties $T[UV] = UV$ if $U$ occurs before $V$, $VU$ if $V$ occurs after $U$; and $T'[UV] = UV$ if $U$ occurs after $V$, $T'[UV] = VU$ if $U$ occurs before $V$. Then, you can write the reduction as
$$W → W_{AC} = \sum_{a,c} T'\left[P_a {P'}_c\right] W T\left[P_a {P'}_c\right].$$
The generalization to more than $2$ quantum events should be fairly obvious, by now.
When this is generalized to field theory, each operator is no longer associated with a specific time, but with a specific space-time point. The Born rule then applies to a finite set of quantum events situated over a point cloud in a compact region of space-time. If the events correspond to the measurements of $𝐀 = A_1, A_2, …, A_n$, then the transition is $W → W_𝐀 = \sum_𝐚 T'\left[P_𝐚\right] W T\left[P_𝐚\right]$, where I'm writing the projections more concisely as $P_𝐚 = P_{a_1} … P_{a_n}$.
In effect, the Born rule then introduces a space-time evolution of the state, even though time is removed out of the picture by making operators (instead of wave vectors) dynamic. If you treat all of space-time as being populated by a (possibly infinite) point cloud, each point associated with a quantum event, then each state $W$ will be associated with a partition of this point cloud into a “before set” and “after set”, with the property that none of the points in the after set can have a future time-like or null curve that leads to any of the points in the “before set”. The ones in the before set are the ones that state may be considered to have already undergone a Born reduction for, while the ones in the after set are those which it has not undergone such a reduction.
Then an effective Born evolution can be defined that produces a transition from any two states whose partitions agree on all but a finite number of points in the point cloud, whenever the before set of the one state (the “later” state) contains the before set of the other state (the “earlier” state). Then the Born rule is applied by taking the quantum events (and their associated operators) that the two states disagree on being before or after. The transition then goes from the earlier state to the later state by applying the reduction that's just been described.
So, in total: this is the closest you get to a straightforward translation of the Born Rule from the Schrödinger Picture to the Heisenberg Picture. The formalism this most closely resembles is what's known as Consistent Histories (https://en.wikipedia.org/wiki/Consistent_histories); whose math is similar. So, you might be able to cannibalize some of their formulae and apply them here, after stripping out all the extra small print they attach to them.
An interesting related reference that I found a few days before this On Quantum Theory treats the Born rule in a similar fashion, but also brings POVM's into the picture. Sections 3 and 6 are where the issues are dealt with.
