Franchise specifications prohibit card surcharges. Small shops and supermarkets all pay card processing fees (they might be a little higher for small shops, who therefore can't absorb them).
Certain cards have what is called "mandatory positive authorization required" flags stored on the chip, stripe, or both. (If you see "application ID" or "AID" A0000032010 on a card receipt, then you have such a card.) These cards might be marked "Electronic use only", or "Visa Electron", or have the numbers printed flat onto the card rather than embossed, or none of the above. Cards with this feature may only be accepted when, before processing the transaction, the merchant has obtained positive authorization from the bank that the cardholder has sufficient funds/limit available to cover the transaction. The usual way of doing this when one is not on a train is for the card terminal to automatically contact the merchant's acquiring bank which will send a request through to the cardholder's bank confirming there are funds available. This can't happen with Avantix Mobile machines as they are not connected to anything. Therefore, such cards will automatically decline.
Because sometimes the mandatory positive authorization flag is stored on the chip and not the stripe, sometimes cards will work when swiped but not when chip-read. For this and other reasons, you are not allowed to swipe a chip card unless you get specific phone authorization to do so and a six-digit code. If you do anyway, the bank is likely to reverse the payment and charge the TOC an admin fee.
None of this will vary by the equipment used, as the setting is stored on the card and the Thyron, Gemalto, and Dionne apparatus should all work exactly the same.
Lloyds TSB green Visa debit cards present a particular problem. Lloyds offers an add-on to its current accounts called "Control" costing £10 a month, in return for which it will guarantee that you will never ever be charged any unauthorized overdraft fees. If you have the add-on, a transaction that takes you below £0 (or your limit) will not work: any direct debits or standing orders will fail, cheques will bounce, and debit card transactions will be declined. Because offline debit card transactions might take you over your limit unexpectedly, they set the cards to reject them as default. However, this add-on can be added or removed by phoning up, and the debit card will not be replaced. To allow for this, they have to issue all debit cards with this restriction already in place. Hopefully that clarifies why Lloyds TSB cards present a problem. Bank of Scotland started to issue these cards in September as well.
Back in the day merchants used to get what is called a "floor limit", whereby their bank would guarantee payment of up to a certain amount (£100 perhaps) for any credit card transaction, even if the accountholder was over the limit. Floor limits started to get phased out in 2001 and now it is mainly only airlines that have them. (Restaurants, hotels, and car hire firms have a variant on floor limits whereby they will still get paid up to 20% more than the amount they authorize to allow for customers leaving tips, having incidental spend, or causing damage respectively.)
As to what TOCs can do about this, installing more TVMs will be a start!