Combining some sheets into another sheet

Untested (ie coded on the fly) but perhaps:

Sub Collate()
Dim ws As Worksheet, wsFinal As Worksheet, rngCopy As Range, xlCalc As xlCalculation
On Error GoTo ExitPoint
With Application
    xlCalc = .Calculation
    .Calculation = xlCalculationManual
    .ScreenUpdating = False
    .EnableEvents = False
End With
Set wsFinal = Sheets("FCombined")
With wsFinal
    .Range(.Cells(2, "A"), .Cells(.Rows.Count, "A")).Resize(,13).Clear
End With
For Each ws In ThisWorkbook.Worksheets
    Select Case UCase(ws.Name)
        Case "F1", "F2", "F3", "F4"
            With ws
                Set rngCopy = .Range(.Cells(2, "A"), .Cells(.Rows.Count, "A").End(xlUp)).Resize(, 13)
            End With
            With wsFinal
                .Cells(.Rows.Count, "A").End(xlUp).Offset(1).Resize(rngCopy.Rows.Count, rngCopy.Columns.Count) = rngCopy.Value
            End With
    End Select
Next ws
ExitPoint:
Set wsFinal = Nothing
With Application
    .Calculation = xlCalc
    .ScreenUpdating = True
    .EnableEvents = True
End With
End Sub

the above assumes

a) final sheet has header row

b) final sheet should be purged with each collation

c) values only copy


EDIT: 12:14 - missing comma in the .Clear line at beginning of routine - did say it was untested