Compare column names and copy data from several sheets only where column exists else blank

I'm not understanding the requirements. Could you give some examples of which data should be copied and the logic. I don't see any columns with the same names as the sheet names for the three sheets. Please clarify.

Steve:

Give this a try:

Sub Core()
    Dim w1 As Worksheet
    Dim w2 As Worksheet
    Dim w3 As Worksheet
    Dim w4 As Worksheet
    Set w1 = Sheets("STPPMD")
    Set w2 = Sheets("ATPMD")
    Set w3 = Sheets("SBMD")
    Set w4 = Sheets("Core")
    Dim lr1 As Long
    lr1 = w1.Range("B" & Rows.Count).End(xlUp).Row
    Dim lr2 As Long
    lr2 = w2.Range("B" & Rows.Count).End(xlUp).Row
    Dim lr3 As Long
    lr3 = w3.Range("B" & Rows.Count).End(xlUp).Row
    Dim lr4 As Long
    Dim i As Long
    Application.ScreenUpdating = False
    
    w1.Activate
    For i = 2 To lr1
    w1.Activate
    lr4 = w4.Range("C" & Rows.Count).End(xlUp).Row
    If Range("B" & i) <> "" Then
    Range("B" & i & ":C" & i).Copy
    w4.Activate
    w4.Range("C" & lr4 + 1).Select
    Selection.PasteSpecial Paste:=xlPasteValues
    End If
    Next i
    Application.CutCopyMode = False
    
    w2.Activate
    For i = 2 To lr2
    w2.Activate
    lr4 = w4.Range("C" & Rows.Count).End(xlUp).Row
    If Range("F" & i) <> "" Then
    Range("F" & i & ":G" & i).Copy
    w4.Activate
    w4.Range("C" & lr4 + 1).Select
    Selection.PasteSpecial Paste:=xlPasteValues
    End If
    Next i
    Application.CutCopyMode = False
    
    w3.Activate
    For i = 2 To lr3
    w3.Activate
    lr4 = w4.Range("C" & Rows.Count).End(xlUp).Row
    If Range("E" & i) <> "" Then
    Range("E" & i & ":F" & i).Copy
    w4.Activate
    w4.Range("C" & lr4 + 1).Select
    Selection.PasteSpecial Paste:=xlPasteValues
    End If
    Next i
    Application.CutCopyMode = False
    
    Application.ScreenUpdating = True
    MsgBox ("Completed")
    
End Sub

Attached is my copy of your workbook with my code. It worked for me. Scroll down the Core page. I wonder if you have some data in that page that makes the last row be something other than row 2.