Ahh, Chase scenarios; those are often tricky. If you have any experience with 4th Edition though, they become less so. True, that 4th edition and 5th edition rules and dynamics are quite different, but there is much that can be sampled and converted.
On page 252 of the 5E DMG (Dungeon Master Guide) you will find the rules on 'Chases' , and this leaves much to be desired as you found out while attempting to use them for your graveyard chase. The roll tables are neat, until you realize they can not apply to every scenario, or environment.
There is nothing else that is 5E SPECIFIC that I can offer you, except my 5E conversion of the 'Non Combat Encounter' system from 4E. So what do you do? You pick an instance or 2 for failure. The party get's too far behind the Githyanki, or the Githyanki reach their objective. Then you create hazards your party might face while chasing down Gith on a moving train with several carts to it. The most important part of it, is to make a list of possible important skill checks and then have the party pick from that list for them to focus on through the course of the chase (how many they pick is dependent upon what the CR of the Chase Encounter is), and they roll those skill checks and the dice decides if they pass or fail.
Example
You notice a Githyanki is attempting to cast Misty Step! (Arcana check to see if anyone proficient in the skill notices) If the roll is higher than a proper DC for their level , they notice and can attempt to counter spell or otherwise stop the Gith from teleporting.
The Githyanki begin to sprint ahead of you! (Atheltics check to see if members of the party can keep up with them) If they fail the check, the party falls behind 1 level, if they succeed then the Athletic member keeps up and leads the party ahead.
A Githyanki launches a Firebolt at you! (Acrobatics check to see if you evade the cantrip.) You can also use random acrobatics checks for them jumping from train car to train car.
These are just a few 'chase situations' that you can apply a Skill Check to and have the party try and get past it by choosing who's the best at each skill and dividing the checks among the party. The best thing about the 'Non Combat Encounter' mechanic is that you can assign exp to these like normal encounters even though they are not engaging in combat where they kill things.+
A good and solid answer was given already:
Modern firearms and grenades appear on page 268 of the Dungeon Master's Guide.
Please upvote that answer if you find it useful, I'm just copying it to comply with the guideline that every answer should stand on it's own even if the others get changed or removed.
However, a paragraph like this has been in all similar editions of D&D (2e, 3.5e and now 5e). I have tried that in the past but that little paragraph actually opens up a whole lot of other connected questions. For example how is armor treated if firearms are supposed to pierce them without problem. A lot of stuff that we take for granted in todays world simply has no rules. Not even conversions. D&D has no rules for chasing each other on horse drawn carts, so there is no easy conversion for modern cars either. The bottom line is, while this is an easy answer, it does not make for a good game conversion, it only leaves you with hundreds of similar questions that don't have answers in the DMG or any other official D&D book. You will probably not be satisfied with your game experience if you take this answer and start playing.
So let me challenge the frame of your question. You want to play a walking dead style campaign with modern weapons. What you have as a tool to do so right now is D&D. And you run into problems because D&D was never meant to do that. There is no modern setting in D&D, it aims very specifically at sword & sorcery, dragons and heroes. No guns, no electricity, no medicine, no internet, but instead you got guys casting spells left and right.
Your question is a bit like having a car, and wanting to have a good sailing trip on the lake with your buddy. That's great. But you run into problems because your car isn't suited for it and now you are asking how to raise a sail on your car because you haven't found the mast yet. Well, there is no mast. It's not meant to do that.
While I'm sure a lot of people will be eager to sell you a mast and a sail and something to waterproof your whole car, let's look outside of the question for a second. How about getting a boat?
Looking around this site for a system that supports a modern day zombie apocalypse, there are threads here, however, they tend to be opinionated and not a good fit for our format.
What would probably work best is to find your local store and just ask them. If it's a good store, they will present you some systems that fit your needs and leave you at a quiet table where you can browse all of them to decide.
To sum it up: get a system that suits your needs. There are plenty out there. Go grab one and have fun.
Best Answer
Chases
Check Chapter 8 of the DMG, the Chases section.
It defines how to begin and end a chase, as well as random complications for both elements. You can easily adapt the urban complications to your setting.
If needed, Waterdeep: Dragon Heist and Baldur's Gate: Descent into Avernus have tables with Chase Complications, where you can take ideas from.
Vehicle Proficiency
Reward players with Vehicle Proficiency, a skill that is (from my experience) rarely useful, and should be emphasized here.
Use speeds based on your vehicle
The Mounts table in Mounts & Vehicles has speeds for different animals pulling your vehicles. Use these to track distances between quarry and pursuers, based on the Chase rules.
Combat
If one of your players is busy handling the vehicle, the others can use their turns to assist him somehow (Gust of Wind on the ship's sails) or to hinder pursuers (Entangle the road behind). I would adjudicate based on their suggestions that vehicles have a small boost/delay, are easier/harder to control, or have more/less complications per round.
The Ghosts of Saltmarsh subreddit has some good threads on ship combat, which you can adapt.
Appendix A of the Ghosts of Saltmarsh module has ship statistics, I'm now sure how adaptable they would be to your setting. Appendix B of Baldur's Gate: Descent into Avernus contains land vehicle statistics.