How can I connect the local minimum-value points between a series of curves by dotted lines?

Question

I have a set of data obtained by NDSolve. After I plot it, I want to indicate some feature of the curves. For example, I want to link the local minimum-value points between these curves by dotted line to highlight the track. First, I import the data:

data1 = Import["C:\\Users\\...\\Desktop\\data1.dat"];
data2 = Import["C:\\Users\\...\\Desktop\\data2.dat"];
data3 = Import["C:\\Users\\...\\Desktop\\data3.dat"];

then, plot them

g1 = ListLinePlot[data1];
g2 = ListLinePlot[data2];
g3 = ListLinePlot[data3];
Show[g1, g2, g3, PlotRange -> All]

I get the following plot

What I want to do is to link the local min-value point by dashed line. Is there any simply method by Mathematica to do this job? Should I make use of FindMinimum and FindMinValue, but I don't know how to combine them. The objective I want to obtain is as follows which is got by hand:).

Yes, as answered by @Oska, whose code can handle many cases. But fail to treat the following case in which there are several local minimum-value points in a single curve. This is the my_data.

Using Oska's code

data1 = Import["C:\\...\\plot1_4sqrt(2)pi.dat"];
data2 = Import["C:\\...\\plot2_4sqrt(2)pi.dat"];
data3 = Import["C:\\...\\plot3_4sqrt(2)pi.dat"];

mins = Function[d, 
d[[#]] & /@ (Sort[
   First@First@
       Position[d, #] & /@ (RankedMin[Last /@ d, #] & /@ {1, 
       2})])] /@ {data1, data2, data3};
ListLinePlot[{data1, data2, data3}, 
Epilog -> {Arrowheads[0.02], 
Arrow /@ (Thread@{(First /@ mins), 
    RotateLeft@(First /@ mins)})[[;; -2]], 
Arrow /@ (Thread@{(Last /@ mins), 
    RotateLeft@(Last /@ mins)})[[;; -2]]}
]

I will get

As can be seen, the code fails to find the local min-value points on the first curve. What I try to get is as follow. I am confusing that why the code can find the local min-value point of the other two curves but fails to the first curve?

Answer 1

I will let you play with the style but here is a beginning:

data1 = Import["~/Downloads/data/data1.dat"];
data2 = Import["~/Downloads/data/data2.dat"];
data3 = Import["~/Downloads/data/data3.dat"];
mins = Function[d, 
   d[[#]] & /@ (Sort[
      First@First@Position[d, #] & /@ (RankedMin[Last /@ d, #] & /@ {1, 2})])] /@ 
   {data1, data2, data3}

ListLinePlot[{data1, data2, data3}, 
 Epilog -> {Red, Line[First /@ mins], Line[Last /@ mins]}]

Regarding the series of arrows:

ListLinePlot[{data1, data2, data3}, 
 Epilog -> {Red, 
   Arrow /@ (Thread@{(First /@ mins), RotateLeft@(First /@ mins)})[[;; -2]], 
   Arrow /@ (Thread@{(Last /@ mins), RotateLeft@(Last /@ mins)})[[;; -2]]},]