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JavaScript (Node.js), 131129 bytes

a=>[...a+''].map(t=>a.map((l,y)=>l.map((v,x)=>(L=a[y+1]L=a[y+1]||[],d=(L||[])[X=x]d=L[X=x],r=l[x+1],v>r?r>=d||(L=l,X++):v>d||(L=l),l[x]=L[X],L[X]=v)))) 

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JavaScript (Node.js), 131 bytes

a=>[...a+''].map(t=>a.map((l,y)=>l.map((v,x)=>(L=a[y+1],d=(L||[])[X=x],r=l[x+1],v>r?r>=d||(L=l,X++):v>d||(L=l),l[x]=L[X],L[X]=v)))) 

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JavaScript (Node.js), 129 bytes

a=>[...a+''].map(t=>a.map((l,y)=>l.map((v,x)=>(L=a[y+1]||[],d=L[X=x],r=l[x+1],v>r?r>=d||(L=l,X++):v>d||(L=l),l[x]=L[X],L[X]=v)))) 

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Source Link
tsh
  • 36.2k
  • 2
  • 36
  • 133

JavaScript (Node.js), 131 bytes

a=>[...a+''].map(t=>a.map((l,y)=>l.map((v,x)=>(L=a[y+1],d=(L||[])[X=x],r=l[x+1],v>r?r>=d||(L=l,X++):v>d||(L=l),l[x]=L[X],L[X]=v)))) 

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