Here's info about those two Battalions from the Liverpool Museum document they sent me....
THE 1/9TH
1/9th Battalion T.F.
4.8.14  
57 Everton Road, Liverpool: South Lancs. Bde. West Lancs. Div. 
13.8.14 
to Edinburgh-Brigade was on Forth Defences. 
Oct. 1914 
to Tunbridge Wells. 
13.3.15 
landed at Havre and to 2nd Bde. 1st Div. 
12.11.15 
to 3rd Bde. 1st Div. 
7.1.16 
to 165th Bde. 55th Div. 
1.2.18 
to 172nd Bde. 57th Div. Absorbing 2/9th Bn. 
11.11.18 
172nd Bde. 57th Div. France; Hellemmes, east of Lille.
THE 13th..
13th (Service) Battalion.
Formed Oct. 1914 at Seaforth, Liverpool-K3-to Salisbury Plain; Army Troops 25th Div. 
Nov. 1914 
to Bournemouth in billets. 
Feb. 1915 
to 76th Bde. 25th Div. 
May 1915 
to Aldershot. 
27.9.15 
landed at Havre. 
15.10.15 
76th Bde. to 3rd Div. 
23.10.15 
to 8th Bde. 3rd Div. 
4.4.16 
to 9th Bde. 3rd Div. 
11.11.18 
9th Bde. 3rd Div. France; Le Grand Sart, east of Le Quesnoy.
Milly