list manipulation - Multidimensional MATLAB conversion

I try to convert this MATLAB code: From:

(useful to compute some prox of some functions):

case 3
    %% 3D field %%
    V          = cat(4, ...
                U.M{1}(1:end-1,:,:) + U.M{1}(2:end,:,:), ...
                U.M{2}(:,1:end-1,:) + U.M{2}(:,2:end,:), ...

                U.M{3}(:,:,1:end-1) + U.M{3}(:,:,2:end));
case 4
    %% 4D field %%
    V          = cat(5, ...
                U.M{1}(1:end-1,:,:,:) + U.M{1}(2:end,:,:,:), ...
                U.M{2}(:,1:end-1,:,:) + U.M{2}(:,2:end,:,:), ...
                U.M{3}(:,:,1:end-1,:) + U.M{3}(:,:,2:end,:),...
                U.M{4}(:,:,:,1:end-1) + U.M{4}(:,:,:,2:end));

"The dimension of U.M{1} is (N+1,N) while that of U.M{2} is (N,N+1) for 2D field". I would like to have the same behavior on Mathematica, without a switch case, on arbitrary dimension. I try to do some mix with Take, Drop, ... Without any success.

rank = 4;
size = 30;
baseDim = ConstantArray[size, rank];
U = Table[Array[Subscript[m, ##] &, ReplacePart[baseDim, k -> size + 1]], {k, 1, rank}];

The idea of the MATLAB code is to concatenate N sub-tensor by taking the begining of and the end of each dimension and sum them up.

Here the '4' stand for the dimension where I would like to catenate, that do the same job as "Join". This is the intuition behind the Mathematica code above. In pure Mathematica for a Tensor with Rank 4 above the special case code can look like:

Join[U[[1]][[2 ;; ,   ;; ,   ;; ,   ;;]] + U[[1]][[;; -2, ;;   , ;;   , ;;   ]],
     U[[2]][[  ;; , 2 ;; ,   ;; ,   ;;]] + U[[2]][[;;   , ;; -2, ;;   , ;;   ]],
     U[[3]][[  ;; ,   ;; , 2 ;; ,   ;;]] + U[[3]][[;;   , ;;   , ;; -2, ;;   ]],

     U[[4]][[  ;; ,   ;; ,   ;; , 2 ;;]] + U[[4]][[;;   , ;;   , ;;   , ;; -2]], 5]

[Edit] The solution based on the answer of @Henrik-Schumacher:

With[{all = ConstantArray[All, rank]},
    Table[U[[k]][[Sequence @@ ReplacePart[all, k -> 2 ;;]]] + U[[k]][[Sequence @@ ReplacePart[all, k -> ;; -2]]], {k, 1, rank}]
]

Answer

Up to the final Join which I don't understand, the following might help:

r = 4;
tmp = Array[Subscript[m, ##] &, ConstantArray[3, r]];


With[{idx = ConstantArray[All, r]},
  Table[
   tmp[[Sequence @@ ReplacePart[idx, i -> 2 ;;]]] + 
    tmp[[Sequence @@ ReplacePart[idx, i -> ;; -2]]],
   {i, 1, r}]
  ]

or

Table[Map[2 MovingAverage[#, {1, 1}] &, tmp, {i - 1}], {i, 1, r}]