Is CNN permutation equivariant?
If you use a single layer CNN, then each vector in the resulting activation maps would be related to the original 3x3 block. However, if you stack multiple CNN layers, you increase the receptive field of each resulting vector, as shown in the image below (taken from here):

After the CNNs, you can certainly compute an LSTM. There are, however, some design decisions you would need to take: disregarding the batch dimension, the LSTM takes as input a sequence of vectors (2D tensor) but you have a 3D tensor (height $\times$ width $\times$ num.channels) so, how are you going to make the 3D tensor fit as input to the LSTM? You could compute the LSTM of columns/rows of the image independently, in forward or reverse direction. You could also &quot;collapse&quot; one dimension (e.g. by averaging or taking the maximum value).