J, 36 bytes

(' ','#.'$~#){~[:+/(-@[|.!.0]$1:)/"2 

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We take input as a list of 2x2 matrices.

• For each of them "2, draw a box of ones 1: of the same shape $ as the right input ], ie, the coords of the bottom right of the box.
• Then rotate that down and to the right with 0 fill |.!.0 until the upper leftmost 1 is in the coords specified by the left arg -@[.
• [:+/ Sum the resulting matrices elementwise. J guarantees they will be the same size by adding 0 fill to the right. This will produce a single matrix with integers 0, 1, 2, 3, etc. We'll use those to index into our ascii chars {~.
• '#.'$~# Repeat the pattern #. cyclically $~ to the input length #, and then prepend a space ' ',.

J, ³⁶ 34 bytes

(' ','#.'$~#){~[:+/(-@]{.-~$1:)/"2 

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-2 bytes thanks to Bubbler

We take input as a list of 2x2 matrices.

• For each of them "2, draw a box of ones 1: of the same shape $ as the element-wise difference between the top-left and bottom-right coords -~.
• Then rotate that down and to the right with 0 fill -@]{. until the upper leftmost 1 is in the coords specified by the left arg -@[.
• [:+/ Sum the resulting matrices element-wise. J guarantees they will be the same size by adding 0 fill to the right. This will produce a single matrix with integers 0, 1, 2, 3, etc. We'll use those to index into our ascii chars {~.
• '#.'$~# Repeat the pattern #. cyclically $~ to the input length #, and then prepend a space ' ',.

J, 36 bytes

(' ','#.'$~#){~[:+/(-@[|.!.0]$1:)/"2 

Try it online!

J, 36 bytes

(' ','#.'$~#){~[:+/(-@[|.!.0]$1:)/"2 

Try it online!

We take input as a list of 2x2 matrices.

• For each of them "2, draw a box of ones 1: of the same shape $ as the right input ], ie, the coords of the bottom right of the box.
• Then rotate that down and to the right with 0 fill |.!.0 until the upper leftmost 1 is in the coords specified by the left arg -@[.
• [:+/ Sum the resulting matrices elementwise. J guarantees they will be the same size by adding 0 fill to the right. This will produce a single matrix with integers 0, 1, 2, 3, etc. We'll use those to index into our ascii chars {~.
• '#.'$~# Repeat the pattern #. cyclically $~ to the input length #, and then prepend a space ' ',.