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Let me add my FDM solution. The spirit of this solution is the same as that of Alex's FDM solution, except that the process is automated with pdetoode and DChange. Since there's nothing new in the following code compared to the previous moving boundary problem, I won't explain much. (Anyway, if you still feel it difficult to follow, feel free to ask in the comment, but please be specific. )

Let me add my FDM solution. The spirit of this solution is the same as that of Alex's FDM solution, except that the process is automated with pdetoode and DChange. Since there's nothing new in the following code compared to the previous moving boundary problems in this site, I won't explain much. (Anyway, if you still feel it difficult to follow, feel free to ask in the comment, but please be specific. )

Something seems to be wrong with moving grid method. Still exploring…

Let me add a solution based on moving grid as shown in

Heat Equation with Mobile Boundary

Moving B.C.s in heat diffusion model

Since there's nothing new in the following code, I won't explain much. Please read the posts linked above carefully. (Anyway, if you still feel it difficult to follow, feel free to ask in the comment, but please be specific. )

I'll use pdetoode for discretization. The initial data is the same as that chosen in Alex's answer:

s0 = 3;
F = t; v0 = x;
With[{v = v[x, t], 
   s = s[t]}, {eq, ic, 
    bc} = {{D[v, t] == D[v, x, x], 
     D[s, t] == -D[v, x] /. x -> s}, {v == v0, s == s0} /. t -> 0, {v == 0 /. x -> 0,
      D[v, x] - 1/s v == F /. x -> s}}];

domain = {0, s[t]};
points = 25;
 grid = Array[# &, points, domain];
 difforder = 2;
(* Definition of pdetoode isn't included in this post,
   please find it in the link above. *)
ptoofunc = pdetoode[v[x, t], t, grid, difforder];
del = #[[2 ;; -2]] &;
freeze = v[i_] :> v@ToString@i(*v[i:Except[_String]]:>v@ToString@InputForm@i*);

ode = {del@ptoofunc@eq[[1]], ptoofunc@eq[[2]]};
odebc = ptoofunc@bc;
odeic = {ic[[1]] // ptoofunc // del, ic[[2]]} /. freeze /. t -> 0;
tend = 1;
{ssol, vsollst} = 
   NDSolveValue[{ode, odeic, odebc} /. freeze, {s, v /@ grid} /. freeze, {t, 0, 
     tend}]; // AbsoluteTiming

vscaledsol = rebuild[vsollst, grid/s[t], 2];
vsol = {x, t} |-> vscaledsol[x/ssol[t], t];
    
reg = DiscretizeRegion@ImplicitRegion[x <= ssol[t], {{t, 0, 1}, {x, 0, 3}}];

DensityPlot[vsol[x, t], {t, x} ∈ reg]

Let me add my FDM solution. The spirit of this solution is the same as that of Alex's FDM solution, except that the process is automated with pdetoode and DChange. Since there's nothing new in the following code compared to the previous moving boundary problem, I won't explain much. (Anyway, if you still feel it difficult to follow, feel free to ask in the comment, but please be specific. )

I've chosen the same initial data as in Alex's answer so we can easily verify the result.

s0 = 3;
F = t; v0 = x;
With[{v = v[x, t], 
   s = s[t]}, {eq, ic, 
    bc} = {{D[v, t] == D[v, x, x], 
     D[s, t] == -D[v, x] /. x -> s}, {v == v0, s == s0} /. t -> 0, {v == 0 /. x -> 0,
      D[v, x] - 1/s v == F /. x -> s}}];
(* Definition of DChange isn't included in this post,
   please find it in the link above. *)       
{neweq, newicmid, newbc} = DChange[{eq, ic, bc}, x/s[t] == ξ, x, ξ, v[x, t]]    
newic = {newicmid[[1]] /. t -> 0 /. s[0] -> s0, newicmid[[2]]};

With[{ic = newic, eq = neweq, bc = newbc},
 domain = {0, 1}; points = 25; grid = Array[# &, points, domain]; difforder = 2;
  (* Definition of pdetoode isn't included in this post,
     please find it in the link above. *)
  ptoofunc = pdetoode[v[ξ, t], t, grid, difforder];
  del = #[[2 ;; -2]] &;
   
  ode = {del@ptoofunc@eq[[1]], ptoofunc@eq[[2]]};
  odebc = ptoofunc@bc;
  odeic = {ic[[1]] // ptoofunc // del, ic[[2]]} /. t -> 0;
  tend = 1;
  {ssol, vsollst} = 
   NDSolveValue[{ode, odeic, odebc}, {s, v /@ grid}, {t, 0, tend}]; // AbsoluteTiming]

vscaledsol = rebuild[vsollst, grid, 2];
vsol = {x, t} |-> vscaledsol[x/ssol[t], t];
    
reg = DiscretizeRegion@ImplicitRegion[x <= ssol[t], {{t, 0, 1}, {x, 0, 3}}];

DensityPlot[vsol[x, t], {t, x} ∈ reg]

Let me add a solution based on moving grid as shown in

Heat Equation with Mobile Boundary

Moving B.C.s in heat diffusion model

Since there's nothing new in the following code, I won't explain much. Please read the posts linked above carefully. (Anyway, if you still feel it difficult to follow, feel free to ask in the comment, but please be specific. )

I'll use pdetoode for discretization. The initial data is the same as that chosen in Alex's answer:

s0 = 3;
F = t; v0 = x;
With[{v = v[x, t], 
   s = s[t]}, {eq, ic, 
    bc} = {{D[v, t] == D[v, x, x], 
     D[s, t] == -D[v, x] /. x -> s}, {v == v0, s == s0} /. t -> 0, {v == 0 /. x -> 0,
      D[v, x] - 1/s v == F /. x -> s}}];

domain = {0, s[t]};
points = 25;
grid = Array[# &, points, domain];
difforder = 2;
(* Definition of pdetoode isn't included in this post,
   please find it in the link above. *)
ptoofunc = pdetoode[v[x, t], t, grid, difforder];
del = #[[2 ;; -2]] &;
freeze = v[i_] :> v@ToString@i(*v[i:Except[_String]]:>v@ToString@InputForm@i*);

ode = {del@ptoofunc@eq[[1]], ptoofunc@eq[[2]]};
odebc = ptoofunc@bc;
odeic = {ic[[1]] // ptoofunc // del, ic[[2]]} /. freeze /. t -> 0;
tend = 1;
{ssol, vsollst} = 
   NDSolveValue[{ode, odeic, odebc} /. freeze, {s, v /@ grid} /. freeze, {t, 0, 
     tend}]; // AbsoluteTiming

vscaledsol = rebuild[vsollst, grid/s[t], 2];
vsol = {x, t} |-> vscaledsol[x/ssol[t], t];
    
reg = DiscretizeRegion@ImplicitRegion[x <= ssol[t], {{t, 0, 1}, {x, 0, 3}}];

DensityPlot[vsol[x, t], {t, x} ∈ reg]

Manipulate[
 Plot[vsol[x, t], {x, 0, ssol[t]}, PlotRange -> {{0, s0}, {-1, 5}}], {t, 0, 1}]

Something seems to be wrong with moving grid method. Still exploring…

Let me add a solution based on moving grid as shown in

Heat Equation with Mobile Boundary

Moving B.C.s in heat diffusion model

Since there's nothing new in the following code, I won't explain much. Please read the posts linked above carefully. (Anyway, if you still feel it difficult to follow, feel free to ask in the comment, but please be specific. )

I'll use pdetoode for discretization. The initial data is the same as that chosen in Alex's answer:

s0 = 3;
F = t; v0 = x;
With[{v = v[x, t], 
   s = s[t]}, {eq, ic, 
    bc} = {{D[v, t] == D[v, x, x], 
     D[s, t] == -D[v, x] /. x -> s}, {v == v0, s == s0} /. t -> 0, {v == 0 /. x -> 0,
      D[v, x] - 1/s v == F /. x -> s}}];

domain = {0, s[t]};
points = 25;
grid = Array[# &, points, domain];
difforder = 2;
(* Definition of pdetoode isn't included in this post,
   please find it in the link above. *)
ptoofunc = pdetoode[v[x, t], t, grid, difforder];
del = #[[2 ;; -2]] &;
freeze = v[i_] :> v@ToString@i(*v[i:Except[_String]]:>v@ToString@InputForm@i*);

ode = {del@ptoofunc@eq[[1]], ptoofunc@eq[[2]]};
odebc = ptoofunc@bc;
odeic = {ic[[1]] // ptoofunc // del, ic[[2]]} /. freeze /. t -> 0;
tend = 1;
{ssol, vsollst} = 
   NDSolveValue[{ode, odeic, odebc} /. freeze, {s, v /@ grid} /. freeze, {t, 0, 
     tend}]; // AbsoluteTiming

vscaledsol = rebuild[vsollst, grid/s[t], 2];
vsol = {x, t} |-> vscaledsol[x/ssol[t], t];
    
reg = DiscretizeRegion@ImplicitRegion[x <= ssol[t], {{t, 0, 1}, {x, 0, 3}}];

DensityPlot[vsol[x, t], {t, x} ∈ reg]

Manipulate[
 Plot[vsol[x, t], {x, 0, ssol[t]}, PlotRange -> {{0, s0}, {-1, 5}}], {t, 0, 1}]