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I am going to replace "kind" with "nationality" and "problem" with "traveler", and see what sounds right to me. Now, we get

What nationality of travelers is being processed?

This sounds fine to me grammatically, but here the sentence implies that you are processing the nationality rather than the travelers. Assuming you're processing the travelers, the sentence is incorrect.

What nationality of travelers are being processed?

Sounds fine to me grammatically, too. And here, since you're processing the travelers, there is actual subject-verb agreement. Note that if you said

What nationality of traveler is being processed?

you would implicitly be talking about only one traveler.

Now, let's consider:

What nationalities of travelers are being processed?

This sounds fine, too. But note there is a difference in the meaning. In the first two questions, the questioner was assuming that you only processed one nationality of travelers at a time. Here, you're processing more than one nationality. Since nationalities and travelers are both plural, the verb is would not work here.

*What nationalities of traveler is being processed.

This sounds wrong to me. Here, there is clearly more than one traveler being processed (because nationalities is plural). You must use are here. I'd say nationalities here is acting somewhat like a collective noun.

So, in your original question, since the patents are being issued rather than the kinds, I would say that the correct phrasing is:

What kind of patents are being issued?

Note that if you were talking to a lawyer in a patent law firm where everybody was a specialist, you would ask:

What kind of patents is your specialty?

because his specialty is a kind and not patents.

Finally, since there is probably more than one kind of patent currently being issued, possibly an even better phrasing (depending on context) would be:

What kinds of patents are being issued.

Since the patents are being issued rather than the kinds, I would say that the correct phrasing is:

What kind of patents are being issued?

Note that if you were talking to a lawyer in a patent law firm where everybody was a specialist, you could ask:

What kind of patents is your specialty?

because his specialty is a kind and not patents.

Finally, since there is probably more than one kind of patent currently being issued, possibly an even better phrasing (depending on context) would be:

What kinds of patents are being issued.

I am going to replace "kind" with "nationality" and "problem" with "traveler", and see what sounds right to me. Now, we get

What nationality of travelers is being processed?

This sounds fine to me grammatically, but here the sentence implies that you are processing the nationality rather than the travelers. Assuming you're processing the travelers, the sentence is incorrect.

What nationality of travelers are being processed?

Sounds fine to me grammatically, too. And here, since you're processing the travelers, there is actual subject-verb agreement.

What nationalities of travelers are being processed?

This sounds fine, too. But note there is a difference in the meaning. In the first two questions, the questioner was assuming that you only processed one nationality of travelers at a time. Here, you're processing more than one nationality. Since nationalities and travelers are both plural, the verb is would not work here.

*What nationalities of traveler is being processed.

This sounds wrong to me. Here, there is clearly more than one traveler being processed (because nationalities is plural). You must use are here. I'd say nationalities here is acting somewhat like a collective noun.

So, in your original question, since the patents are being issued rather than the kinds, I would say that the correct phrasing is:

What kind of patents are being issued.

Note the similarity to

A lot of patents are being issued.

However, since there is probably more than one kind of patent currently being issued, possibly an even better phrasing (depending on context) would be:

What kinds of patents are being issued.

I am going to replace "kind" with "nationality" and "problem" with "traveler", and see what sounds right to me. Now, we get

What nationality of travelers is being processed?

This sounds fine to me grammatically, but here the sentence implies that you are processing the nationality rather than the travelers. Assuming you're processing the travelers, the sentence is incorrect.

What nationality of travelers are being processed?

Sounds fine to me grammatically, too. And here, since you're processing the travelers, there is actual subject-verb agreement. Note that if you said

What nationality of traveler is being processed?

you would implicitly be talking about only one traveler.

Now, let's consider:

What nationalities of travelers are being processed?

This sounds fine, too. But note there is a difference in the meaning. In the first two questions, the questioner was assuming that you only processed one nationality of travelers at a time. Here, you're processing more than one nationality. Since nationalities and travelers are both plural, the verb is would not work here.

*What nationalities of traveler is being processed.

This sounds wrong to me. Here, there is clearly more than one traveler being processed (because nationalities is plural). You must use are here. I'd say nationalities here is acting somewhat like a collective noun.

So, in your original question, since the patents are being issued rather than the kinds, I would say that the correct phrasing is:

What kind of patents are being issued?

Note that if you were talking to a lawyer in a patent law firm where everybody was a specialist, you would ask:

What kind of patents is your specialty?

because his specialty is a kind and not patents.

Finally, since there is probably more than one kind of patent currently being issued, possibly an even better phrasing (depending on context) would be:

What kinds of patents are being issued.

I am going to replace "kind" with "nationality" and "problem" with "traveler", and see what sounds right to me. Now, we get

What nationality of travelers is being processed?

This sounds fine to me grammatically, but here the sentence implies that you are processing the nationality rather than the travelers. Assuming you're processing the travelers, the sentence is incorrect.

What nationality of travelers are being processed?

Sounds fine to me grammatically, too. And here, since you're processing the travelers, there is actual subject-verb agreement.

What nationalities of travelers are being processed?

This sounds fine, too. But note there is a difference in the meaning. In the first two questions, the questioner was assuming that you only processed one nationality of travelers at a time. Here, you're processing more than one nationality. Since nationalities and travelers are both plural, the verb is would not work here.

*What nationalities of traveler is being processed.

This sounds wrong to me. Here, there is clearly more than one traveler being processed (because nationalities is plural). You must use are here.

So, in your original question, since the patents are being issued rather than the kinds, I would say that the correct phrasing is:

What kind of patents are being issued.

However, since there is probably more than one kind of patent currently being issued, an even better phrasing would be:

What kinds of patents are being issued.

I am going to replace "kind" with "nationality" and "problem" with "traveler", and see what sounds right to me. Now, we get

What nationality of travelers is being processed?

This sounds fine to me grammatically, but here the sentence implies that you are processing the nationality rather than the travelers. Assuming you're processing the travelers, the sentence is incorrect.

What nationality of travelers are being processed?

Sounds fine to me grammatically, too. And here, since you're processing the travelers, there is actual subject-verb agreement.

What nationalities of travelers are being processed?

This sounds fine, too. But note there is a difference in the meaning. In the first two questions, the questioner was assuming that you only processed one nationality of travelers at a time. Here, you're processing more than one nationality. Since nationalities and travelers are both plural, the verb is would not work here.

*What nationalities of traveler is being processed.

This sounds wrong to me. Here, there is clearly more than one traveler being processed (because nationalities is plural). You must use are here. I'd say nationalities here is acting somewhat like a collective noun.

So, in your original question, since the patents are being issued rather than the kinds, I would say that the correct phrasing is:

What kind of patents are being issued.

Note the similarity to

A lot of patents are being issued.

However, since there is probably more than one kind of patent currently being issued, possibly an even better phrasing (depending on context) would be:

What kinds of patents are being issued.