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tms=Table[ArrayPlot[Last/@TuringMachine[r,{{1,10},Table[0,21]},10]],{r,0,4095}];
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Show[#,ImageSize40]&/@tms;
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%
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Revised Code

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getTuringMachineStateEvolutionFunction[state_String,rules_List]:=ToString/@(Last[TuringMachine[#,ToExpression[state],1]]&/@rules)
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getTuringMachineStateEventFunction[state_String,rules_List]:=Module[{evolution=ToString/@(Last[TuringMachine[#,ToExpression[state],1]]&/@rules)},​​Module[{length=StringLength[Last[Last[System`Dump`showStringDiff[state,#]]]]},{StringDrop[state,-length]StringDrop[#,-length],StringDrop[state,-length],{"",StringTake[state,-length]}}]&/@evolution]
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getTuringMachineStateEventFunction[states_List,rules_List]:=Catenate[getTuringMachineStateEventFunction[#,rules]&/@states]
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getTuringMachineEventApplicationFunction[{(input_String|input_Symbol)output_String,(input_String|input_Symbol),{prefix_String,suffix_String}}]:=StringJoin[prefix,output,suffix]
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getTuringMachineElementShifts[shiftedFragment_String,{input:{inputPrefix_String,inputSuffix_String},output:{outputPrefix_String,outputSuffix_String}}]:=Table[Rule@@({StringPart[shiftedFragment,position],{StringJoin[#1,StringTake[shiftedFragment,;;position-1]],StringJoin[StringTake[shiftedFragment,position+1;;],#2]}}&@@@{input,output}),{position,StringLength[shiftedFragment]}]
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getTuringMachineElementShifts[{(input_String|input_Symbol)output_String,(input_String|input_Symbol),{prefix_String,suffix_String}}]:=Join[getTuringMachineElementShifts[prefix,{{"",StringJoin[input,suffix]},{"",StringJoin[output,suffix]}}],getTuringMachineElementShifts[suffix,{{StringJoin[prefix,input],""},{StringJoin[prefix,output],""}}]]
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getTuringMachineElements[{substring_String,{prefix_String,suffix_String}}]:={StringPart[substring,#],{StringJoin[prefix,StringTake[substring,;;#-1]],StringJoin[StringTake[substring,#+1;;],suffix]}}&/@Range[StringLength[substring]]
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getTuringMachineEventDecompositionFunction[event:{(input_String|input_Symbol)output_String,(input_String|input_Symbol),{prefix_String,suffix_String}}]:=Join[{getTuringMachineElementShifts[event]},getTuringMachineElements[{#,{prefix,suffix}}]&/@{input,output}]
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turingMachineStateRenderingFunction:=(Inset[Framed[Style[RulePlot[TuringMachine[2506(*needsasampleofruletype*)],ToExpression[#2],0,MeshAll,FrameFalse],Hue[0.62,1,0.48]],BackgroundDirective[Opacity[0.2],Hue[0.62,0.45,0.87]],FrameMargins{{2,2},{0,0}},RoundingRadius0,FrameStyleDirective[Opacity[0.5],Hue[0.62,0.52,0.82]]],#1,Center,#3]&)
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getTuringMachineEventRenderingForm[function_,{(input_String|input_Symbol)output_String,(input_String|input_Symbol),{prefix_String,suffix_String}}]:=Row[{Column[{function[ArrayPlot[{Last[ToExpression[StringJoin[prefix,input,suffix]]]}]],function[ArrayPlot[{Last[ToExpression[StringJoin[prefix,output,suffix]]]}]]},Center,0]}]
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turingMachineEventRenderingFunction:=(If[First[First[#2]]===Null,​​Inset[Framed[Style[getTuringMachineEventRenderingForm[Identity,#2],Hue[0.09,1,0.32]],BackgroundDirective[Opacity[0.7],RGBColor[0.259,0.576,1]],FrameMargins{{2,2},{0,0}},RoundingRadius0,FrameStyleDirective[Opacity[0.4],Hue[0.09,1,0.91]]],#1,Center,#3],​​Inset[Framed[Style[getTuringMachineEventRenderingForm[Identity,#2],Hue[0.09,1,0.32]],BackgroundDirective[Opacity[0.7],Hue[0.14,0.34,1]],FrameMargins{{2,2},{0,0}},RoundingRadius0,FrameStyleDirective[Opacity[0.4],Hue[0.09,1,0.91]]],#1,Center,#3]]&)
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MultiwayTuringMachine[rules_List,initialCondition_,stepCount_Integer,rest___]:=ResourceFunction["MultiwaySystem"][<|"StateEvolutionFunction"(getTuringMachineStateEvolutionFunction[#,rules]&),"StateEquivalenceFunction"SameQ,"StateEventFunction"(getTuringMachineStateEventFunction[#,rules]&),"EventDecompositionFunction"getTuringMachineEventDecompositionFunction,"EventApplicationFunction"getTuringMachineEventApplicationFunction,"SystemType""TuringMachine","EventSelectionFunction"Identity|>,{ToString[initialCondition]},stepCount,rest,"StateRenderingFunction"turingMachineStateRenderingFunction,"EventRenderingFunction"turingMachineEventRenderingFunction]

Running code

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RulePlot[TuringMachine[2506],{{1,3,0},{0,0,0,0,0}},0,MeshAll,FrameFalse]
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MultiwayTuringMachine[{2506,3506},{{1,3,0},{0,0,0,0,0}},2,"EvolutionEventsGraph",VertexSize1]
»
{{{1, 3, 0}, {0, 0, 0{{2, 2, -1}, {0, 0, 1,{{1, 3, 0}, {0, 0, 0,{,, 0, 0}}}}
»
{{{1, 3, 0}, {0, 0, 0{{2, 4, 1}, {0, 0, 1,{{1, 3, 0}, {0, 0, 0,{,, 0, 0}}}}
»
{{{2, 2, -1}, {0, 0{{1, 1, -2}, {0, 1,{{2, 2, -1}, {0, 0,{,, 1, 0, 0}}}}
»
{{{2, 4, 1}, {0, 0, 1, 0{{1, 3, 0}, {0, 0, 1, 1,{{2, 4, 1}, {0, 0, 1, 0,{,, 0}}}}
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MultiwayTuringMachine[Range[1000],{{1,3,0},{0,0,0,0,0}},2,"StatesGraph",VertexSize4]
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MultiwayTuringMachine[Range[1000],{{1,3,0},{0,0,0,0,0}},2,"StatesGraph",VertexSize{.4,.1}]
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The head cannot have yet colored the end squares in the tape...
In 3 steps, the head can only have reached 3 squares to color them....
Distance between the most distant 1s can be at most 3 ; also there is a correlation between head position and tape configuration
Number of possible states after t steps / AKA number of nodes in the graph:

Pruning of cases

Only need to consider outcome from the current state+color; all rules with different outcomes for different state+colors are irrelevant.
I.e. just need a representative sample of the 4096 possible rules.....

What is the causal graph like for all these rules?

https://www.wolframscience.com/nks/p489--time-and-causal-networks/
Note the presence of many branchial event horizons......

Higher States, Colors