Matrix times Vector where the elements are vectors

Whats the correct operation to calculate the "product" of matrix $$A$$ of the size $$M \times L$$

$$A= \begin{bmatrix} \vec{A}_{1,1} & \vec{A}_{1,2} \\ \vec{A}_{2,1} & \vec{A}_{2,2} \\ \vec{A}_{3,1} & \vec{A}_{3,2} \\ \end{bmatrix}$$ where the elements are vectors of the form $$\vec{A}_{1,1} = \begin{bmatrix}A_{1,1}^1, A_{1,1}^2,...,A_{1,1}^N \end{bmatrix}$$ (where the entries like $$A_{1,1}^1$$ are a complex number)

with a vector $$b$$ of the size M $$b=\begin{bmatrix} \vec{b_1}\\ \vec{b_2}\\ \vec{b_3} \end{bmatrix}$$ where the elements of $$b$$ are vectors too for example $$\vec{b_1}=\begin{bmatrix} b_1^1, b_1^2, ... ,b_1^N \end{bmatrix}$$ (where the entries like $$b_{1}^1$$ are a complex number)

such that

$$A^T \bullet b = \begin{bmatrix} \vec{A}_{1,1} \odot \vec{b}_1 + \vec{A}_{2,1} \odot \vec{b}_2 + \vec{A}_{3,1} \odot \vec{b}_3\\ \vec{A}_{1,2} \odot \vec{b}_1 + \vec{A}_{2,2} \odot \vec{b}_2 + \vec{A}_{3,3} \odot \vec{b}_3 \end{bmatrix}$$

where $$\odot$$ denotes the hadamard product (elementwise multiplication of two vectors of the same length).

My questions are:

1. Is there such an operation?
2. What is the correct mathematical notation to define such an operation
3. Is this somehow easier to write? For example as a Tensor - Matrix operation where $$A \in \mathbb{C}^{M \times L \times N}$$ and $$b \in \mathbb{C}^{M \times N}$$?

$$\def\o{{\tt1}}$$The dimension $$M$$ must be factorable such that the $$b$$ vector can be arranged as a block vector of $$p$$ partitions each of length $$N$$, i.e. $$M=pN.\;$$ In your example $$\;p=3,\;N=M/p$$.
Using all-ones vectors $$({\rm eg}\;\o_p\in{\mathbb R}^{p})$$, Kronecker products $$(\otimes)$$, the identity matrix $$\,\left(I_N\in{\mathbb R}^{N\times N}\right)$$, and vectorization (aka column stacking), the desired operation can be written as \eqalign{ &{\rm vec}\Big(\big(\o_p\otimes I_N\big)^T\left(A\odot b\o_L^T\right)\Big)\\ &{\rm vec}\Big(\big(\o_p\otimes I_N\big)^T\,{\rm Diag}(b)\,A\Big) \\ }