Follow the dots

APL, 63 bytes

{⍺[(⊂1 1){×≢⍵:⍺,(V/⍵)∇⍵/⍨~V←(+=⌊/)+/¨2*⍨⍺-⍵⋄⍺}1↓(,⍵='.')/,⍳⍴⍵]}

This is a function that takes two character matrices (they must be matrices), the character grid as its left argument and the dots grid as its right argument. The matrix of dots may be smaller than the matrix of characters.

Explanation:

• (,⍵='.')/,⍳⍴⍵: get the positions of the dots, in row-column order
• 1↓: drop the first one (it is known to be at 1 1)
• (⊂1 1){...}: starting from 1 1, run the following function, which follows the path (its left argument is its current position, its right argument are unvisited positions). It works by selecting the nearest unvisited dot each time. (If the assumptions from the question hold, this is correct.)
  • ×≢⍵:: if there are still unvisited positions:
    • +/¨2*⍨⍺-⍵: find the Manhattan distance between each position and the current position
    • V←(+=⌊/): for each position, see if its distance is equal to the smallest distance, and store this in V.
    • ⍵/⍨~: select all positions for which this is not the case, these are the fields to visit next
    • (V/⍵): find the position for which it is the case, this will be the next field
    • ∇: run the function again with these new arguments
    • ⍺,: the result is the current position, followed by the result of doing this for the rest of the list
  • ⋄⍺: otherwise, just return the current position and stop (it's the last one)
• ⍺[...]: for each position, select the corresponding element in the character grid.

Test cases:

      f←{⍺[(⊂1 1){×≢⍵:⍺,(V/⍵)∇⍵/⍨~V←(+=⌊/)+/¨2*⍨⍺-⍵⋄⍺}1↓(,⍵='.')/,⍳⍴⍵]}
      ⍝ example
      g0  ← 4 5⍴'ABCDEFGHIJKLMNOPQRST'
      d0  ← 4 5⍴'..  . . .. . .  ... '
      ⍝ test case 1
      g1  ← 6 7⍴'ABCACBWDEFGHUQXLUSDQZASUKWXIWUKOAIMAIAIOUP'
      d1a ← 6 7⍴'..      ...      .   ....   .      .      '
      d1b ← 6 7⍴'.......      ....   .. ..  ..     ........'
      ⍝ test case 2
      g2  ← 2 2⍴'ABCD'
      d2a ← 1 1⍴'.'
      d2b ← 1 2⍴'..'
      d2c ← 2 2⍴'. ..'
      ⍝ test case 3
      g3  ← 1 1⍴'A'
      d3  ← 1 1⍴'.'

      g0 f d0
ABGLQRSNIJE
      (⊂g1) f¨ d1a d1b
┌────────────┬─────────────────────────┐
│ABEFGSKUSAWA│ABCACBWQZIMPUOIAIAWAXLUUK│
└────────────┴─────────────────────────┘
      (⊂g2) f¨ d2a d2b d2c
┌─┬──┬───┐
│A│AB│ACD│
└─┴──┴───┘
      g3 f d3
A

JavaScript (ES6), 122 bytes

c=>g=>c[l=~c.search`
`,i=p=0]+[...g].map(_=>i|!p?c[i=(d=n=>g[i-n-p?i-n:c]>" "&&(p=i)-n)(1)||d(-1)||d(l)||d(-l)]:"").join``

Explanation

Takes the grids as multiline strings.

Feels like a decent attempt but I ran out of time while golfing so it could probably be improved.

var solution =

c=>g=>
  c[                            // add the starting letter to the output
    l=~c.search`
`,                              // l = line length
    i=p=0                       // i = current index, p = previous index
  ]+
  [...g].map(_=>                // loop
    i|!p?                       // if we have not finished yet
      c[i=                      // find the next index and return it's letter
        (d=n=>                  // d = function to check for a dot at offset n
          g[
            i-n-p?i-n           // if i - n != p, get the character at index i - n
            :c                  // else get index "c" (will return undefined, not a dot)
          ]>" "                 // if the character is a dot
          &&(p=i)-n             // set p to i and return i - n
        )
        (1)||d(-1)||d(l)||d(-l) // search for the next adjacent dot
      ]
    :""                         // if we have finished, return no letter
  )
  .join``                       // output all the returned letters

<textarea id="Characters" rows="6" cols="30">ABCABCW
DEFGHUQ
XLUSDQZ
ASUKWXI
WUKOAIM
AIAIOUP</textarea>
<textarea id="Grid" rows="6" cols="30">.......
      .
...   .
. ..  .
.     .
.......</textarea><br />
<button onclick="result.textContent=solution(Characters.value)(Grid.value)">Go</button>
<pre id="result"></pre>