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I need to create several graph for different values of a parameter $h$.

The code is the following

pdf1[x_] = PDF[NormalDistribution[1, 1], x]

pdf[x_] = PDF[NormalDistribution[0, 1], x]

cdf1[x_] = CDF[NormalDistribution[1, 1], x]

cdf[x_] = CDF[NormalDistribution[0, 1], x]

h = 0; l = .48; SeedRandom[1900];

f1[x_] := pdf1[x]/pdf[x] (h + (1 - h) 2 cdf1[x])/(h + (1 - h) 2 cdf[x])

f2[x_] :=  pdf1[x]/pdf[x] (h + (1 - h) 2 (1 - cdf1[x] + cdf1[a]))/(h + (1 - h) 2 (1 - cdf[x] + cdf[a]))

f3[x_] := pdf1[x]/pdf[x] (h + (1 - h) 2 (cdf1[x] - cdf1[b] + cdf1[a]))/(h + (1 - h) 2 (cdf[x] - cdf[b] + cdf[a]))

f4[x_] := pdf1[x]/pdf[x] (h + (1 - h) 2 (1 - cdf1[x] + cdf1[c] - cdf1[b] + cdf1[a]))/(h + (1 - h) 2 (1 - cdf[x] + cdf[c] - cdf[b] + 

    cdf[a]))

amin = NArgMin[{f2[a], f2[a] > l}, a], amax = NArgMax[{f1[a], f1[a] < l}, a]



btab = Interpolation@Table[{a, NArgMax[{f3[b], f3[b] < l}, b]}, {a, amin, amax, .02}];



ctab = Interpolation[Flatten[Table[{{a, b}, 

  Quiet@NArgMax[{f3[c], f3[c] < l, b < c}, c]}, {a, amin, amax, .02}, {b, a, btab[a], .02}], 1], InterpolationOrder -> 1];



 Quiet@RegionPlot3D[amin < a < amax && a < b < btab[a] && b < c < ctab[a, b], {a, amin, amax}, {b, amin, btab[amin]}, {c, amin, ctab[amin, btab[amin]]}, AxesLabel -> {a, b, c}, 

LabelStyle -> {Black, Bold, Medium}, BoxRatios -> Automatic, ImageSize -> Large, PlotPoints -> 500, Mesh -> None]

I would like to generate several graphs for different values of $h$, say for $h$ between 0 and 1, with $dh$=0.01.
How can I do this?

The explicit code needed to answer your question is

pdf1[x_] = PDF[NormalDistribution[1, 1], x];

pdf[x_] = PDF[NormalDistribution[0, 1], x];

cdf1[x_] = CDF[NormalDistribution[1, 1], x];

cdf[x_] = CDF[NormalDistribution[0, 1], x];



f1[x_] := pdf1[x]/pdf[x] (h + (1 - h) 2 cdf1[x])/(h + (1 - h) 2 cdf[x])

f2[x_] := pdf1[x]/pdf[x] (h + (1 - h) 2 (1 - cdf1[x] + cdf1[a]))/(h + (1 - h) 

    2 (1 - cdf[x] + cdf[a]))

f3[x_] := pdf1[x]/pdf[x] (h + (1 - h) 2 (cdf1[x] - cdf1[b] + cdf1[a]))/(h + (1 - h) 

    2 (cdf[x] - cdf[b] + cdf[a]))

f4[x_] := pdf1[x]/pdf[x] (h + (1 - h) 2 (1 - cdf1[x] + cdf1[c] - cdf1[b] + 

    cdf1[a]))/(h + (1 - h) 2 (1 - cdf[x] + cdf[c] - cdf[b] + cdf[a]))



l = .48;



Column@Table[

    amin = NArgMin[{f2[a], f2[a] > l}, a]; amax = NArgMax[{f1[a], f1[a] < l}, a];

    btab = Interpolation@Table[{a, NArgMax[{f3[b], f3[b] < l}, b]}, 

        {a, amin, amax, (amax - amin)/20}];

    fc[a0_, b0_] := Quiet@NArgMax[{f3[c], f3[c] < l, b < c} /. {a -> a0, b -> b0}, c];

    Plot3D[{b, fc[a, b]}, {a, amin, amax}, {b, a, btab[a]}, AxesLabel -> {a, b, c}, 

        PlotLabel -> StringForm["h = ``", h], LabelStyle -> {Black, Bold, 15}, 

        BoxRatios -> Automatic, ImageSize -> Large, Mesh -> None, 

        PlotStyle -> Opacity[.5], PlotPoints -> 10, MaxRecursion -> 0], 

{h, 0, .98, .49}]

Note that I used the more accurate second addendum to my earlier answer for a single plot. I also used Plot3D instead of RegionPlot3D, because the latter is very slow here. Even the calculation above is slow, of course.

Something like

GraphicsColumn[Table[Plot[some expression with h in it],{h,0,1,0.01}]]

perhaps.

Here is an example in which I use Manipulate to plot two graphs for different parameter values. This example may help you to adapt your code in a similar manner.

Clear[alfa, newalfa, a1, a2, x, s, chn];

Manipulate[

 SeedRandom[s];

 alfa = RandomReal[1, 20];

 newalfa = alfa*(1 + chn);

 Manipulate[

  Plot[{Sin[a1 x], Sin[a2 x]}, {x, 0, 10}],

  Row[{Control[{a1, alfa, Animator, AnimationRunning -> False}]}],

  Row[{Control[{a2, newalfa, Animator, AnimationRunning -> False}]}]

  ],

 {{s, 1, "s"}, 1, 100, 1},

 {{chn, 0, "change"}, -0.2, 0.2, 0.02}

 ]

This code yields the following graphs.

In all honesty this does nothing more than @John Doty suggested, but it illustrates my point in an earlier comment.

Define

h = Range[1, 10, 0.5]

a = 1

b = 1

f[a_, b_, h_, x_] := PDF[NormalDistribution[h, b + a], x]

g[a_, b_, h_, x_] := PDF[InverseGaussianDistribution[a + h, b], x]

Then

Do[{hh = h[[j]], Print[Plot[{f[a, b, hh, x], g[a, b, hh, x]}, {x, -5, 5}, 

PlotLegends -> "Expressions"]]}, {j, 1, Length[h]}]

It will print 20 PDF plots given a and b evaluated over range x, for each value of h. From what I understand in the OP, the part of the code below the definition of f4 can go into the brackets above, with the plot put inside a Print. Also, it looks like the brackets around amin, amax are reduntant, but I may be wrong. The operations inside the Do loop can be as complex as needed, so the calculation of the tables can be put in directly. Unfortunately I was not able to run the code in the OP and see what I was supposed to get.