mjhjmtu

That is what I have so far:

documentclass[tikz,border=3.14mm]standalone
begindocument
usetikzlibraryintersections,decorations

begintikzpicture[thick,
 dot/.style = 
 draw,
 fill = black,
 circle,
 inner sep = 0pt,
 minimum size = 2pt
]

 draw[smooth cycle, tension=0.6] plot coordinates(-1,-.4) (1.1,-.6) (.9,.5) (-1.1,.5) node at (1,.6) $Omega$;
 draw[blue] (-1.1,.5) .. controls (-.5,-.2) and (.5,0) .. coordinate[dot,pos=0.3,black] (A) coordinate[dot,pos=0.5,black] (B) coordinate[dot,pos=0.7,black] (C) coordinate[pos=0.92,black] (D) (1.1,-.6);
 draw[gray] (A) circle (.3);
 draw[gray] (B) circle (.375) node[black,anchor=north] $x_0$;
 node[anchor=south,gray] at ([yshift=10]B) $gamma(barD)$;
 draw[gray] (C) circle (.3);
 node[blue,anchor=south west] at (D) $S$;

endtikzpicture

enddocument

which looks like

Now, I want to add two dashed lines at each side of S going from the beginning and end of S touching the circles exactly such that each circle stays within.

Something like:

UPDATE: Using tangents, see e.g. this nice answer.

documentclass[tikz,border=3.14mm]standalone
usetikzlibrarycalc
begindocument

begintikzpicture[thick,
 dot/.style = 
 draw,
 fill = black,
 circle,
 inner sep = 0pt,
 minimum size = 2pt
]

 draw[smooth cycle, tension=0.6] plot coordinates(-1,-.4) (1.1,-.6) (.9,.5) (-1.1,.5) node at (1,.6) $Omega$;
 draw[blue] (-1.1,.5) coordinate(start) .. controls (-.5,-.2) and (.5,0) ..
 coordinate[dot,pos=0.3,black] (A) coordinate[dot,pos=0.5,black] (B)
 node[black,anchor=north] $x_0$ coordinate[dot,pos=0.7,black] (C) coordinate[pos=0.92,black] (D) (1.1,-.6)
 coordinate(end);
 % see https://tex.stackexchange.com/a/76226/121799
 pgfmathsetmacrorsmall0.6
 pgfmathsetmacrorbig0.75
 node [draw=gray,circle,minimum size=rsmall*1cm] (cA) at (A);
 node [draw=gray,circle,minimum size=rbig*1cm] (cB) at (B);
 node[anchor=south,gray] at ([yshift=10]B) $gamma(barD)$;
 node [draw=gray,circle,minimum size=rsmall*1cm] (cC) at (C);
 node[blue,anchor=south west] at (D) $S$;
 beginscope[overlay] % see https://tex.stackexchange.com/a/76226/121799
 coordinate (cAB) at (barycentric cs:B=-rsmall,A=rbig);
 coordinate (cCB) at (barycentric cs:B=-rsmall,C=rbig);
 endscope
 foreach Y in 1,2
 foreach X in A,C
 path (tangent cs:node=cX,point=(cX B),solution=Y) coordinate(pBX-Y)
 (tangent cs:node=cB,point=(cX B),solution=Y) coordinate(pX B-Y);
 path (tangent cs:node=cA,point=(start),solution=Y) coordinate(pstartA-Y)
 (tangent cs:node=cC,point=(end),solution=Y) coordinate(pendC-Y); 
 draw[red,dashed] plot[smooth] coordinates (start) (pstartA-2) (pBA-2) (pAB-2) (pCB-1) (pBC-1)
 (pendC-1) (end);
 draw[red,dashed] plot[smooth] coordinates (start) (pstartA-1) (pBA-1) (pAB-1) (pCB-2) (pBC-2)
 (pendC-2) (end);
endtikzpicture
enddocument

OLDER ANSWER: You started drawing this with Bezier curves (which I may not necessarily have done), so perhaps the best way is to continue this strategy. Yet I think it helps to make the circles nodes such that you can easily access the points on their peripheries.

documentclass[tikz,border=3.14mm]standalone
begindocument

begintikzpicture[thick,
 dot/.style = 
 draw,
 fill = black,
 circle,
 inner sep = 0pt,
 minimum size = 2pt
]

 draw[smooth cycle, tension=0.6] plot coordinates(-1,-.4) (1.1,-.6) (.9,.5) (-1.1,.5) node at (1,.6) $Omega$;
 draw[blue] (-1.1,.5) .. controls (-.5,-.2) and (.5,0) ..
 coordinate[dot,pos=0.3,black] (A) coordinate[dot,pos=0.5,black] (B)
 node[black,anchor=north] $x_0$ coordinate[dot,pos=0.7,black] (C) coordinate[pos=0.92,black] (D) (1.1,-.6);
 node [draw=gray,circle,minimum size=0.6cm] (cA) at (A);
 node [draw=gray,circle,minimum size=0.75cm] (cB) at (B);
 node[anchor=south,gray] at ([yshift=10]B) $gamma(barD)$;
 node [draw=gray,circle,minimum size=0.6cm] (cC) at (C);
 node[blue,anchor=south west] at (D) $S$;
 draw[red,dashed] (-1.1,0.5) to[out=-20,in=190] (cA.110)
 to[out=10,in=170] (cB.90) to[out=-10,in=170] (cC.80) to[out=-10,in=150] (1.1,-.6); 
 draw[red,dashed] (-1.1,0.5) to[out=-50,in=150] (cA.250)
 to[out=-20,in=180] (cB.270) to[out=00,in=190] (cC.280) to[out=10,in=180] (1.1,-.6); 
endtikzpicture
enddocument

An empirical solution:

documentclass[tikz,border=3.14mm]standalone
begindocument
begintikzpicture[thin,
 dot/.style = 
 draw,
 fill = black,
 circle,
 inner sep = 0pt,
 minimum size = 2pt
]

 draw[smooth cycle, tension=0.6] plot coordinates(-1,-.4) (1.1,-.6) (.9,.5) (-1.1,.5) node at (1,.6) $Omega$;
 draw[blue] (-1.1,.5)coordinate(s1) .. controls (-.5,-.2) and (.5,0) .. coordinate[dot,pos=0.3,black] (A) coordinate[dot,pos=0.5,black] (B) node[black,below]$x_0$ coordinate[dot,pos=0.7,black] (C) coordinate[pos=0.92,black] (D) (1.1,-.6)coordinate(s2);
 draw (A) node[draw,minimum size=0.52cm,circle] (C1);
 draw (B) node[draw,minimum size=0.65cm,circle] (C2);
 draw (C) node[draw,minimum size=0.52cm,circle] (C3);
 node[blue,anchor=south west] at (D) $S$;
 node[anchor=south,gray] at ([yshift=10]B) $gamma(barD)$;
 draw[dashed,smooth, tension=0.5] plot coordinates(s1) (C1.87) (C2.85) (C3.60) (s2);
 draw[dashed,smooth, tension=0.5] plot coordinates(s2) (C3.267) (C2.270) (C1.240) (s1); 
endtikzpicture
enddocument