How to replace motor mount

In the past, I have used floor jack, in area of motor mount, raise engine just enough to give you some slack at the motor mount, to remove and install motor mount. Be careful where you put the jack, don't damage anything and be safe. Safety is your responsibility. I can't see anything from here.
I don't know which motor mount your talking about, exactly?

Transmission mounts are worn out and most likely your motor mounts too. Pretty straight forward job to replace but sometimes the bolts holding the motor mounts will be stuck requiring an impact wrench. Problem is that there is no room to get the impact near these bolts making a lift necessary.

I have no video of the job just written instructuions.

Park the vehicle on level ground and shut off the engine. Raise the front end of the Escort off the ground using the floor jack. Lower the vehicle onto jack stands placed under the frame to support the car.

2Place the jack under the oil pan and raise the lift plate to the bottom of the oil pan. Remove the engine mount bolt using the 1/2-inch socket wrench. The engine mount bolt is in the center of the mount fitting in clear view.

Slowly raise the engine using the jack until the weight of the engine is removed from the engine mount. Pull the engine mount from the fitting and place the new mount in the fitting.

4Lower the engine slowly and insert the bolt into the mounting hole. Tighten the mount bolt to 67.9 foot-pounds using the torque wrench.

5Remove the bolt from the other engine mount the same way as the first one. Pull the mount from the fitting and insert the new one. Replace the bolt to the center of the mount fitting. Lower the engine to the frame and tighten the mount bolt to 67.9 foot-pounds.

6Remove the jack from under the Escort. Raise the car off the jack stands, remove the stands and lower the car to the ground

It is normal for the engine to move a little, but if it moves quite a bit tou could have a broken motor mount. If a mount is broken you will usually hear a thump in the vehicle either when put into gear or upon acceleration or deceleration.

possible but more likely issue with valve body in trans.is it a 5 speed auto?does it get worse the hotter the trans gets?the mounts are a common failure on these but generally when they fail you get a big thump when you initially hit this gas when pulling away from being stopped.

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The Motor raising up issue is, you may have a worn or broken Engine or Transmission Mount, inspect the mounts by having someone or you, look at the mounts while someone else puts the car in gear. If you see the mounts move excessively, than the mounts need to be replaced. The whining or squealing Fan Belt may be a worn, glazed or loose belt. I would inspect the belt and tighten or replace as needed.

Sounds to be a engine mount. Put the car in in 1st as you would take off. With the brake on. See if the engine is pulling up on that side. Also do these in reverse.Which ever side your engine moves, is most likely the broke engine mount. To replace this, I usually loosen the mount, jack the engine up just enough for the engine to raise to sit on, take the old mount out and install the new. tighten the bolts and release jack slowly. Go through and tighten the bolts again for tightness. The dealer states on a small car the time is 3-4 hours, but I had my old off and new on within 7 min. So it's an easy job.

I believe you have a transmission mount that is bad, possibly and engine mount. When you are shifting the engine will raise and bump the firewall, floorpan.
Good luck

The brakes are the cause of the pulling isue. So get that fixed first. It's caused by uneven wear- one side grabbing better than the other. Find out if it's in the front or rear and replace both calipers (front or rear). Always repalce brakes in a set (front/rear).

The motor mounts were probably just noticed by the mechanic. After 8/9 years the mounts get a bit of play in them and you would feel a thump when accelerating, shifting.... If they were really bad i think that would cause a slight pull, but that would be in an extreme case